## Kerala Syllabus 7th Standard Maths Solutions Chapter 5 Area of a Triangle

You can Download Area of a Triangle Questions and Answers, Activity, Notes, Kerala Syllabus 7th Standard Maths Solutions Chapter 5 to help you revise the complete Syllabus and score more marks in your examinations.

## Kerala State Syllabus 7th Standard Maths Solutions Chapter 5 Area of a Triangle

### Area of a Triangle Text Book Questions and Answers

Dot Math Textbook Page No. 68

In the picture below, the horizontal and vertical dots are one centimetre apart.

What are the areas of the coloured figures below?

In the above picture, draw other figures by joining dots and find out their areas.
Figure 1 is 13 cm,
Figure 2 is 6 cm
Figure 3 is 12 cm.

Explanation:
The area of figure 1 is 7+6 =13 cm,
figure 2 is 2+2+2 = 6 cm,
figure 3 is 3+3+3+3 = 12 cm.

This can be done using the Grid option in GeoGebra. Select the Polygon tool and click at various intersections of the grid lines to draw different figures.

Calculate the area of these. To check the answers, select the Area tool and click within each figure.

Different halves
A rectangle can be halved by cutting vertically or horizontally along the middle:

We can also cut corner to corner to make two triangles of half the area.

What happens when we draw a slanted line through the middle’?

Explanation:
By drawing a slanted line through the middle of the rectangle we will get two quadrilaterals.

Didn’t we get two quadrilaterals of half the area?
Such a quadrilateral with only one pair of parallel sides is called a trapezium.

Parallelogram and rectangle Textbook Page No. 70

How do we compute the area of this parallelogram?

We can compute the area of this parallelogram by multiplying the base of the perpendicular by its height.

Cut out a right angled triangle from the left as shown below.

And place it on the right like this:

Now we get a rectangle of the same area.

Rectangle and triangle
In this picture below, what part of the rectangle is the red triangle?

Here, the red triangle is half part of the rectangle as the rectangle surrounds the triangle.

Rectangle and triangle Textbook Page No. 72
How about cutting the rectangle into two smaller rectangles?

The area of the red triangle in each of these pieces is half the area of that piece. So the sum of their areas is equal to half the area of the original rectangle.
The original red triangle is made up of these two smaller red triangles.
Thus the area of the original red triangle is half the area of the original rectangle.

What about the red triangle above?

Explanation:
When the red triangle in the image above is divided in two, a square and a rectangle are formed.
The square and the rectangle are made up of these two smaller red triangles.
The area of square and rectangle will be the sum of two formed half triangles. The area of original triangle will be the half the area of square and rectangle.

Draw a rectangle in GeoGebra and mark a point on the top side. Select the Polygon tool and draw a triangle as in the problem. Colour it red. Select the Area tool and find the area of the triangle.
Drag the point on top and see what happens to the area.

Here, we have drawn a rectangle in GeoGebra and marked a point on the top side and selected the Polygon tool and drawn a triangle, and selected the area tool.

Rectangles within rectangle
Look at this rectangle:

Do you see any relation between the areas of the red rectangles?
Think for a moment before turning the page for the answer.

Rectangles within rectangle Textbook Page No. 74
The diagonal of the large rectangle divides it into two right angled triangles of equal area. Each of these triangles is made up of the red rectangle within it and two small right angled triangles.

The areas of right angled triangles of the same colour are equal.
So, the area of the red rectangles are also equal.
What if we draw rectangle through some other point on the diagonal?

The areas of right angled triangles with the same colour are equal even if we draw a rectangle via another point on the diagonal.
So, the area of the red rectangles will also be equal.

In GeoGebra, draw a pair of horizontal lines, 8 units apart. On the bottom line, mark two points D and F 4 units apart. Mark a point B on the top line and draw ΔDFB using the Polygon tool. What is the area of this triangle? Check the answer using the Area tool. Now drag B and see what happens to the area.

Square division
Draw a square and mark the mid points of its sides.

Join these to the corners of the square as shown below:

We get a square at the centre:

What part of the original square is this?

Square division Textbook Page No. 76

Cut out a figure like this in paper.

Rearrange the triangular pieces like this:

Now we get five squares of equal size.
So, the small square is $$\frac{1}{5}$$ of the orignal square.

Trapezium Textbook Page No. 77

In the figure ABCD is a rectangle and EFG is a right angled triangle.
What are the areas of the trapeziums AFED, and ECBF?

Explanation:
Given that the figure ABCD is a rectangle and EFG is a right-angled triangle. So the areas of the trapeziums AFED, and ECBF is
Here, the total area will be
= 20 × 10
= 200 sq cm.
The area of the rectangle is ECBG
= EC × BC
= 6 × 10
= 60 sq cm.
The area of the triangle EGF is
= $$\frac{1}{2}$$ × 4 × 10
= 2 × 10
= 20 sq cm.
So the area of the trapezium ECBF is
= area of the rectangle is ECBG + area of the triangle EGF
= 60 + 20
= 80 sq cm.

Halving Textbook Page No. 68

Cut out a paper rectangle 4 centimetres wide and 3 centimetres high.

Draw a line down the middle as below:

Now we have two rectangles. What is the area of each?
To see that it’s half the big rectangle, we need only fold it across.
So the area of a small rectangle is half the area of the large rectangle. That is,
$$\frac{1}{2}$$ × 12 = 6 square centimetres
Can you halve the area in any other way?

The image above shows another method for halving the area.
The diagonal of the rectangle divides it into two triangles of equal area.
Area of triangle will be $$\frac{1}{2}$$×12 = 6 square centimetres.
Therefore, the area of original rectangle will be twice of the triangle , 2×6 = 12 square centimetres.

Another half
Cut out a paper rectangle 10 centimetres wide and 8 centimetres high.

Draw a line from corner to corner as shown.
The rectangle is split into two triangles.
Are their areas equal?
Can we fold and check as before?
Then put one on top of the other.
So, what is the area of each triangle?
The area of a triangle is half the area of the rectangle. That is,
$$\frac{1}{2}$$ × 10 × 8 = 40 sq. cm.
Do you notice anything about the angles of these triangles?

A triangle with a right angle at one corner is called a right-angled triangle.
What is the area of the right-angled triangle shown on the right?

The area of the right-angled triangle is 10 sq cm.

Explanation:
The area of the right-angled triangle is
= $$\frac{1}{2}$$ × 4 × 5
= 2 × 5
= 10 sq cm.

Cut out two right angled triangles like this and place them together as shown below.

What is the area of this rectangle?
The area of the right-angled triangle is half of this, right?
Area of the triangle = $$\frac{1}{2}$$ × 4 × 5
= 10 sq. cm.
In this, 4 and 5 are the lengths of the perpendicular sides of the right-angled triangle.
Thus we have the method to compute the area of a right-angled triangle.

The area of a right-angled triangle is half the product of the perpendicular sides.

Now calculate the area of the figures shown below:

The area of the figures is 30 cm sq, 8.75 cm sq, 13.5 cm sq.

Explanation:
The area of the first figure is
= $$\frac{1}{2}$$ × 10 × 6
= 5 × 6
= 30 cm sq.
The area of the second figure is
= $$\frac{1}{2}$$ × 3.5 × 5
= $$\frac{17.5}{2}$$
= 8.75 cm sq.
The area of the third figure is
= $$\frac{1}{2}$$ × 4.5 × 6
= 4.5 × 3
= 13.5 cm sq.

The total area is 40 cm sq.

Explanation:
The area of the rectangle is length × breadth
= 6 × 5
= 30 cm sq.
And the area of the triangle is
= $$\frac{1}{2}$$ × 2 × 5
= 5 cm sq.
The total area is the area of the rectangle + the area of the triangle
= 30 + 5 + 5
= 40 cm sq.

The total area is 16.5 cm sq.

Explanation:
The area of the rectangle is length × breadth
= 4 × 3
= 12 cm sq.
The area of the first triangle is
= $$\frac{1}{2}$$ × 1 × 3
= $$\frac{3}{2}$$
= 1.5 cm sq.
The area of the second triangle is
= $$\frac{1}{2}$$ × 2 × 3
= 3 cm sq.
The total area is the area of the rectangle + the area of the triangle
= 12 + 1.5 + 3
=16.5 cm sq.

The total area of the triangle is 15 cm sq.

Explanation:
The area of the first triangle is
= $$\frac{1}{2}$$ × 3 × 6
= 3 × 3
= 9 cm sq.
The area of the second triangle is
= $$\frac{1}{2}$$ × 2 × 6
= 6 cm sq.
The total area of the triangle is
= 9 + 6
=15 cm sq.

• The area of a right-angled triangle is 96 square centimeters. One of the perpendicular sides is 16 centimeters long. What is the length of the other?

The length of the other side is 6 sq cm.

Explanation:
Given that the area of a right-angled triangle is 96 square centimeters and one side of the perpendicular sides is 16 centimeters. So the length of the other side is
96 = 16 × L
L = 96 ÷ 16
= 6 sq cm.

• The perpendicular sides of a right-angled triangle are 12 and 15 centimeters long. Another right angled triangle of the same area has one of the perpendicular sides 18 centimeters long. What is the length of the other?

The area of the first right-angled triangle is 90 sq cm and the length of the other side is 10 cm.

Explanation:
Given the perpendicular sides of a right-angled triangle are 12 and 15 centimeters long. So the area of the first triangle is
= $$\frac{1}{2}$$ × 12 × 15
= 90 sq cm.
So the area will be
90 = $$\frac{1}{2}$$ × 18 × h
90 = 9 × h
h = $$\frac{90}{9}$$
= 10 cm.

Other triangles Textbook Page No. 72

Look at this triangle:

No angle of it is right.
How do we calculate the area?
Can we cut it into two right-angled triangles?
Look at the earlier problems you have done.

So to compute the area, what all lengths are to be measured?

Area = ($$\frac{1}{2}$$ × 2 × 3) + ($$\frac{1}{2}$$ × 4 × 3)
= 3 + 6
= 9 sq. cm.
We can calculate the area of any triangle like this.
What is the general method to calculate the area of a triangle?
Look at this triangle:

To find the area, first, draw a perpendicular from the top comer to divide it into two right-angled triangles.

Now some lengths are to be measured.
Let’s write letters for these for the time being.

So, how do we write the area?
The sum of the areas of two right-angled triangles is
($$\frac{1}{2}$$ × x × z) + ($$\frac{1}{2}$$ × y × z)
= $$\frac{1}{2}$$xz + $$\frac{1}{2}$$yz
= $$\frac{1}{2}$$(x + y)z
In this, x + y is the length of the bottom side.

So, how do we compute the area of a triangle?
The area of a triangle is half the product of one side with the perpendicular from the opposite side.

Now compute the area of these figures:

The area of the first triangle is 25 sq cm.
The area of the second triangle is 11 sq cm.

Explanation:
The area of the first triangle is
= $$\frac{1}{2}$$ × 5 × 10
= 5 × 5
= 25 sq cm.
The area of the second triangle is
= $$\frac{1}{2}$$ × 5.5 × 4
= 5.5 × 2
= 11 sq cm.

The area of the first figure is 9 sq cm.
The area of the second figure is 16 sq cm.

Explanation:
The area of the first figure is
= $$\frac{1}{2}$$ × 3 × 6
= 3 × 3
= 9 sq cm.
The height of the second triangle is
= 2+6
= 8 cm and the base is 4 cm.
The area of the second figure is
= $$\frac{1}{2}$$ × 8 × 4
= 4 × 4
= 16 sq cm.

The area of the first figure is 9 sq cm.
The area of the second figure is 16 sq cm.
The area of the third figure is 21 sq cm.

Explanation:
The area of the first figure is
= $$\frac{1}{2}$$ × 3 × 5
= $$\frac{15}{2}$$
= 7.5 sq cm.
The area of the second figure is
= $$\frac{1}{2}$$ × 4 × 5
= 2 × 5
= 10 sq cm.
The area of the third figure is
= $$\frac{1}{2}$$ × 6 × 7
= 3 × 7
= 21 sq cm.
So the total area of the figure is 7.5+10+21 = 38.5 sq cm.

The area of the first figure is 24 sq cm.
The area of the second figure is 24 sq cm.
The area of the rectangle is 120 sq cm.
The total area is 168 sq cm.

Explanation:
The area of the first figure is
= $$\frac{1}{2}$$ × 4 × 12
= 2 × 12
= 24 sq cm.
The area of the second figure is
= $$\frac{1}{2}$$ × 4 × 12
= 2 × 12
= 24 sq cm.
The area of the rectangle is
= l × b
= 10 × 12
= 120 sq cm.
So the total area is
= 24 + 24 + 120
= 168 sq cm.

Another triangle Textbook Page No. 75
Look at this triangle:

How do we compute its area?
How do we draw a perpendicular from A to BC?
How about extending BC to the right’?

Now how do we calculate the area of ΔABC?
We get ΔABC by removing ΔACD from ΔABD.

ΔABD is a right-angled triangle.

Area of ΔABD = $$\frac{1}{2}$$ × BD × AD
ΔACD also is a right-angled triangle.
Area of ΔACD = $$\frac{1}{2}$$ × CD × AD
Now we can find the area of ΔABC
Area of ΔABC = Area of ΔABD – Area of ΔACD
= $$\frac{1}{2}$$ × BD × AD – $$\frac{1}{2}$$ × CD × AD
= $$\frac{1}{2}$$ × (BD – CD) × AD
From the picture,
BD – CD = BC
Thus we have
Area of ΔABC = $$\frac{1}{2}$$ × (BD – CD) × AD
= $$\frac{1}{2}$$ × BC × AD
Measure BC, and AD and compute the area.
Here AD is the height measured from BC.
So for triangles of this kind also, the area is half the product of a side and the height from it.

Look at this triangle:

Compute the area of the triangle by measuring out the needed lengths.
Let the base of triangle be 6cm wide and the height be 4cm long.
Thus, the area of triangle will be $$\frac{1}{2}$$ × 6 × 4 = 12 square cm.

Let’s do it! Textbook Page No. 77

Question 1.
A rectangular plot is 30 metres long and 10 metres wide. Within it, a triangular part is marked for plant¬ing plantain.

(i). What is the area of this part?
Area = 300 m sq.

Explanation:
Given that a rectangular plot is 30 meters long and 10 meters wide. So the area of the land will be 30×10 = 300 m sq.
The area of part 1 is
= 10 × 10
= 100 sq m.
The area of part 2 is
= $$\frac{1}{2}$$ × 4 × 10
= 2 × 10
= 20 sq m.
The area of part 3 is
= $$\frac{1}{2}$$ × 16 × 10
= 8 × 10
= 80 sq m.

(ii). What is the area of the triangular part to the right of the area for plantain?
The area of triangular part to the right of the area for plantain is 20 sq m.

Explanation:
The area of triangular part to the right of the area for plantain is
= $$\frac{1}{2}$$ × 4 × 10
= 2 × 10
= 20 sq m.

(iii). What is an area of the trapezium to the left of the plantain area?
The area of the trapezium to the left of the plantain area is 120 sq m.

Explanation:
The area of the trapezium to the left of the plantain area is
= area of part 1 + area of part 2
= 100 + 20
= 120 sq m.

Question 2.
In ΔABC, the angle at B is right. Its area is 48 square centimeters and the length of BC is 8 centimeters. The side of BC is extended by 6 centimeters to D. What is the area of ΔADC?
The length of AB is 12 cm and the area of ADC is 84 sq cm.

Explanation:
Given that the area of ΔABC is 48 sq cm and the length of BC is 8 cm, so
ΔABC = $$\frac{1}{2}$$ × BC × AB
48 =  $$\frac{1}{2}$$ × 8 × AB
48 = 4 × AB
AB = $$\frac{48}{4}$$
= 12 cm.
And the area of ADC is
ΔADC = $$\frac{1}{2}$$ × AD × AB
= $$\frac{1}{2}$$ × 14 × 12
= 7 × 12
= 84 sq cm.

## Kerala Syllabus 6th Standard Basic Science Solutions Guide

Expert Teachers at HSSLive.Guru has created Kerala Syllabus 6th Standard Basic Science Solutions Guide Pdf Free Download in both English Medium and Malayalam Medium of Chapter wise Questions and Answers, Notes are part of Kerala Syllabus 6th Standard Textbooks Solutions. Here HSSLive.Guru has given SCERT Kerala State Board Syllabus 6th Standard Basic Science Textbooks Solutions Pdf of Kerala Class 6 Part 1 and Part 2.

## Kerala State Syllabus 6th Standard Basic Science Textbooks Solutions

Kerala Syllabus 6th Standard Basic Science Guide

Kerala State Syllabus 6th Standard Basic Science Textbooks Solutions Part 1

• Chapter 1 Caskets of Life
• Chapter 2 The Essence of Change
• Chapter 3 Flower to Flower
• Chapter 4 Along with Motion
• Chapter 5 Food for Health

Kerala State Syllabus 6th Standard Basic Science Textbooks Solutions Part 2

• Chapter 6 Living in Harmony
• Chapter 7 Attraction and Repulsion
• Chapter 8 Moon and Stars
• Chapter 9 Mix and Separate
• Chapter 10 For Shape and Strength

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## Kerala State Syllabus 6th Standard Textbooks Solutions

We hope the given Kerala Syllabus 6th Standard Textbooks Solutions Guide Pdf Free Download all Subjects in both English Medium and Malayalam Medium of Chapter wise Questions and Answers, Notes will help you. If you have any queries regarding SCERT Kerala State Board Syllabus Class 6th Textbooks Answers Guide Pdf of Part 1 and Part 2, drop a comment below and we will get back to you at the earliest.

## Kerala Syllabus 6th Standard Maths Solutions Guide

Expert Teachers at HSSLive.Guru has created Kerala Syllabus 6th Standard Maths Solutions Guide Pdf Free Download in English Medium and Malayalam Medium of Chapter wise Questions and Answers, Notes are part of Kerala Syllabus 6th Standard Textbooks Solutions. Here HSSLive.Guru has given SCERT Kerala State Board Syllabus 6th Standard Maths Textbooks Solutions Pdf of Kerala Class 6 Part 1 and Part 2.

## Kerala State Syllabus 6th Standard Maths Textbooks Solutions

Kerala Syllabus 6th Standard Maths Guide

Kerala State Syllabus 6th Standard Maths Textbooks Solutions Part 1

Kerala State Syllabus 6th Standard Maths Textbooks Solutions Part 2

We hope the given Kerala Syllabus 6th Standard Maths Solutions Guide Pdf Free Download in both English Medium and Malayalam Medium of Chapter wise Questions and Answers, Notes will help you. If you have any queries regarding SCERT Kerala State Board Syllabus Class 6th Maths Textbooks Answers Guide Pdf of Part 1 and Part 2, drop a comment below and we will get back to you at the earliest.

## Kerala State Syllabus 7th Standard Maths Solutions Chapter 6 Square and Square Root

### Square and Square Root Text Book Questions and Answers

Triangular numbers Textbook Page No. 80

See the dots arranged in triangles:

How many dots are there in each?
1, 3, 6
How many dots would be there in the next triangle?
Such numbers as 1, 3, 6, 10, … are called triangular numbers.
The first triangular number is 1.
The second is 1 + 2 = 3.
The third is 1 + 2 + 3 = 6.
What is the tenth triangular number?
From the above logic we need to add 10 numbers sum to obtain the 10th triangular number.
To find the 10th triangular number T10 we need to add 1+ 2+3+4+5+6+7+8+9+10 = 55
Therefore, 10th triangular number T10 is 55.

Squares and triangles Textbook Page No. 81

Look at these pictures:

Each square is divided into two triangles.
Let’s translate this into numbers:
4 = 1 + 3
9 = 3 + 6
16 = 6 + 10
Check whether the same pattern continues. What do we see?
All perfect squares after 1 are the sums of two consecutive triangular numbers.
What is the sum of the seventh and eighth triangular numbers?
Yes adding consecutive triangular numbers gives us perfect squares. So, we get the next pattern 9+16 = 25 which in turn is a perfect square.
7th Triangular number can be obtained as T7 = 1+2+3+4+5+6+7 = 28
8th Triangular number can be obtained as T8 = 1+2+3+4+5+6+7+8 = 36
Sum of Seventh and Eighth Triangular Numbers = T7+T8
= 28+36
= 64
Thus, the sum of 7th, 8th Triangular Numbers is 64

Increase and decrease Textbook Page No. 82

Look at this number pattern:
1 = 1
4 = 1 + 2 + 1
9 = 1 + 2 + 3 + 2 + 1
16 = 1 + 2 + 3 + 4 + 3 + 2 + 1

Can you split some more perfect squares like this?
Writing some more perfect squares we have 25 = 1+2+3+4+5+4+3+2+1
36 = 1+2+3+4+5+6+5+4+3+2+1
49 = 1+2+3+4+5+6+7+6+5+4+3+2+1
64 = 1+2+3+4+5+6+7+8+7+6+5+4+3+2+1

Square difference

We have see that
22 = 12 + (1 + 2)
32 = 22 + (2 + 3)
42 = 32 + (3 + 4) and so on.
We can write these in another manner also:
22 – 12 = 1 + 2
32 – 22 = 2 + 3
42 – 32 = 3 + 4
In general, the difference of the squares of two consecutive natural numbers is their sum. Now look at these:
32 – 12 = 9 – 1 = 8
42 – 22 = 16 – 4 = 12
52 – 32 = 25 – 9 = 16
What is the relation between the difference of the squares of alternative natural numbers and their sum?
32 – 12 = (3+1) x 2 = 8
42 – 22 = (4+2) x 2 = 12
52 – 32 =  = (5+3) x 2 = 16

The difference between squares of two alternate natural numbers is always even i.e. twice the sum of two numbers that are squared.

Project Textbook Page No. 84

Last digit
Look at the last digit of squares of natural numbers from 1 to 10:
1, 4, 9, 6, 5, 6, 9, 4, 1, 0
Now, look at the last digits of squares of natural numbers from 11 to 20.
Do we have the same pattern?
Let’s look at another thing: Does any perfect square end in 2?
Which are the digits which do not occur at the end of perfect squares?
Is 2637 then a perfect square?
Last digits of squares of natural numbers from 11 to 20 are 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. Yes they have the same pattern
As we observe the digits 2, 3, 7, 8 doesn’t occur at the end of perfect squares.
As we know 2637 ends with digit 7 at the end it is not a perfect square.

To decide that a number is not a perfect square, we need only look at the last digit.
Can we decide that a number is a perfect square from its last digit alone?
Yes, we can decide if a number is perfect square or not by seeing the last digit alone as you can see above.

Rectangle and square

Look at this picture.

Dots in a rectangle.
Can you rearrange the dots to make another rectangle?
Can you rearrange the dots to make a square? Start like this:

How many more are needed to make a square?

How many dots were there in the original rectangle? How many in this square?
What do we see here?
42 = (3 × 5) + 1
Can we do this for all rectangular arrangements?
The numbers here are 3, 4, 5.
So, for this trick to work, what should be the relation between the number of dots in each row and column of the rectangle?
We can write this in numbers as
22 = (1 × 3) + 1
32 = (2 × 4) + 1
42 = (3 × 5) + 1
Try to continue this
52 = (4 x 6) +1
62 = (5 x 7) +1
72 = (6 x 8) +1
82 = (7 x 9) +1
92 = (8 x 10)+1

Square root of a perfect square

784 is a perfect square. What is its square root? 784 is between the perfect squares 400 and 900; and we know that their square roots are 20 and 30. So $$\sqrt{784}$$ is between 20 and 30. Since last digit of 784 is 4, its square root should have 2 or 8 as the last digit. So $$\sqrt{784}$$ is either 22 or 28.
784 is near to 900 than 400. So $$\sqrt{784}$$ must be 28. Now calculate 282 and check.
Given that 1369, 2116, 2209 are perfect squares, find their square roots like this.

Project Textbook Page No. 87

Digit sum

16 is a perfect square and the sum of its digits is 7.
The next perfect square 25 also has digit sum 7.
The digit sum of 36 is 9.
The sum of the digits of the next perfect square 49 is 13. If we add the digits again, the sum is 4. Find the sum of the digit sums (reduced to a single digit number) of perfect squares starting from 1.
Do you see any pattern?
Is 3324 is perfect square?
Given number is 3324
Now Split the number and add each number 3 + 3 + 2 + 4  = 12
As the result is more than 1 number we should add it again 1 + 2 = 3
All possible numbers that are perfect square have roots of either 1, 4, 7, 9
As 3 is not in the list 3324 is not a Perfect Square.

Rows and columns Textbook Page No. 80

Look this picture:

Dots in rows and columns make a rectangle.
How many dots in all?
Did you count the dots one by one?
Can you make other rectangles with 24 dots?
Is any one of these a square?
How many more dots do we need to make a square? Can you remove some dots and make a square? How many?
Can you remove some dots and make a square? How many?
Numbers which can be arranged in squares are called square numbers.
Do you see anything special about of the number of dots making a square?
There are total 4×6=24 dots in all.
No, the dots were counted by multiplying the number of dots in rows and number of dots in column.
None of these is a square.
We need more 12 dots to make a square, 24+12=36 or 62.
Yes, we can remove some dots and make a square.
We need to remove 8 dots to make a square, 24-8=16 or 42.
The number of dots making a square is the sqaure of that number.

Squares

What are the ways in which we can write 36 as the product of two numbers?
2 × 18, 3 × 12, 4 × 9
We can also write
36 = 6 × 6
And we have seen that it can also be shortened as 36 = 62.
36 is 6 multiplied by 6 itself; that is, the second power of 6.
There is another name for this:
36 is the square of 6.
Then what is the square of 5?
What is the square of $$\frac{1}{2}$$?
Square of 5 = 52 = 25
$$\frac{1}{2}$$ × $$\frac{1}{2}$$ = $$\frac{1}{4}$$

Perfect squares

1, 4, 9, 16,… are the squares of the natural numbers. They are called perfect squares.
What is the perfect square after 16?
Why is 20 not a perfect square?
Prime Factorization of 20 can be written as 22 x 51
As 5 is not in pair 20 is not a perfect square.
The next perfect square after 16 is 25.

Let us look at the succession of perfect squares in another way.
To reach 4 from 1, we must add 3.
To reach 9 from 4?
We can state these as
4 – 1 = 3
9 – 4 = 5
16 – 9 = 7
All these differences are odd numbers, right?
So, the difference of two consecutive perfect squares is an odd number.
Let’s write this as,
4=1 + 3
9 = 4 + 5 = 1 + 3 + 5
16 = 9 + 7 = 1 + 3 + 5 + 7
What do we see here?
When we add consecutive odd numbers starting from 1, we get the perfect squares.
This can be seen from these pictures also.

Can you write down the squares of natural numbers upto 20, by adding odd numbers? You can proceed like this
12 = 1
22 = 1 + 3 = 4
32 = 4 + 5 = 9
42 = 9 + 7 = 16

What is the relation between the number of consecutive odd numbers from 1 and their sum?
What is the sum of 30 consecutive odd numbers starting from 1?
Let us assume the arithmetic series 1, 3, 5, 7, 9, 11, 13…..
we know the formula for sum of arithmetic series Sn = n/2[2a+(n-1)d]
Here d = 2
first term a = 1
Substituting the inputs we have Sn= 30/2[2×1+(30-1)2]
= 30/2[2+58]
= 30/2[60]
= 900
Therefore, the sum of 30 consecutive odd numbers starting from 1 is 900.

Tricks with ten Textbook Page No. 82

The square of 10 is 100. What is the square of 100?
In the square of 1000, how many zeros are there after 1?
What about the square of 10000?
What happens to the number of zeros on squaring?
So how do we spot the perfect squares among 10, 100, 1000, 10000 and so on?
Is one lakh a perfect square?
Now find out the squares of 20, 200 and 2000.
Is 400000000 a perfect square?
What if we put in one more zero?
Square of a number containing x zeros will become 2 times number of zeros.
1 lakh is not a perfect square as the number of zeros is not even.
Ten Lakhs is a perfect square as it comes with 6 zeros that are even.
20 = 20 x 20 = 400
200 = 200 x 200 = 40000
2000 = 2000 x 2000 = 4000000
400000000 is a perfect square as the number of zeros 8 is even.

• Find out the squares of these numbers.

• 30
302 = 30 x 30
= 900
The square of 30 is 900.

• 400
4002 = 400 x 400
= 160000
The square of 400 is 160000

• 7000
70002 = 7000 x 7000
= 49000000
The Square of 7000 is 49000000

• 6 × 1025
(6 × 1025)= (6 × 1025) x (6 × 1025)
= 36 x (1025)2
= 36 x 1050

• Find out the perfect squares among these numbers.

• 2500
The given number 2500 can be written as 502
Hence the 2500 is a perfect square number.

• 36000
We cannot write the given number 36000 as square of two equal numbers. Hence it is not a perfect square.

• 1500
We cannot write the given number 1500 as square of two equal numbers. Hence it is not a perfect square.

• 9 × 107
We cannot write the given number 9 × 107 as square of two equal numbers. Hence it is not a perfect square.

• 16 × 1024
The given number 2500 can be written as (4x 1012)2
Hence the given number 16 × 1024 is a perfect square.

Next square

What is the square of 21?
Wait a bit before you start multiplying.
The square of 20 is 400, isn’t it? So to get the square of 21, we need only add an odd number.
Which odd number?
Let’s start from the beginning. We can write
22 = 12 + 3 = 12 + (1 + 2)
32 = 22 + 5 = 22 + (2 + 3)
42 = 32 + 7 = 32 + (3 + 4)
52 = 42 + 9 = 42 + (4 + 5)
and so on. Continuing like this, how do we write 212?
212 = 202 + (20 + 21)
That is,
212 = 400 + 41 = 441
Now we can continue as before with
222 = 441 +43 = 484
and so on.
How do we find out the square of 101?
1002 = 10000
100 + 101 = 201
So, 1012 = 10000 + 201 = 10201

• Find out the squares of these numbers using the above idea.
• 51
Using the above process we write the 512 = 502+(50+51)
= 2500+(50+51)
= 2500+101
= 2601

• 61
It can be written as 612 = 602+(60+61)
= 3600+(60+61)
= 3600+121
= 3721

• 121
The given number is written as 1212 = 1202+(120+121)
= 14400+(120+121)
= 14400+241
= 14641

• 1001
It is written as 10012 = 10002+(1000+1001)
= 10002+(1000+1001)
= 1000000+2001
= 1002001

• Compute the squares of natural numbers from 90 to 100.
902
= 90 x 90
=8100
912
= 91 x 91
= 8281
922
= 92 x 92
= 8464
932
= 93 x 93
= 8649
942
= 94 x 94
= 8836
952
= 95 x 95
= 8025
962
= 96 x96
= 8928
972
= 97 x 97
= 9409
982
= 98 x 98
= 9310
992
= 99 x 99
= 9801
1002
= 100 x 100
= 10,000

Fraction squares Textbook Page No. 83

A fraction multiplied by itself is also a square.
What is the square of $$\frac{3}{4}$$ ?
($$\frac{3}{4}$$)2 = $$\frac{3}{4}$$ × $$\frac{3}{4}$$ = $$\frac{3 \times 3}{4 \times 4}$$ = $$\frac{9}{16}$$
That is,
($$\frac{3}{4}$$)2 = $$\frac{9}{16}$$ = $$\frac{3^{2}}{4^{2}}$$
So to square a fraction, we need only square the numerator and denominator separately.

Now do these problems without pen and paper.
• Find out the squares of these numbers.
• $$\frac{2}{3}$$
$$\frac{2}{3}$$2  = $$\frac{2}{3}$$*$$\frac{2}{3}$$
= $$\frac{2×2}{3×3}$$
= $$\frac{4}{9}$$

• $$\frac{1}{5}$$
$$\frac{1}{5}$$2  = $$\frac{1}{5}$$*$$\frac{1}{5}$$
= $$\frac{1×1}{5×5}$$
= $$\frac{1}{25}$$

• $$\frac{7}{3}$$
$$\frac{7}{3}$$2  = $$\frac{7}{3}$$*$$\frac{7}{3}$$
= $$\frac{7×7}{3×3}$$
= $$\frac{49}{9}$$

• 1$$\frac{1}{2}$$
1$$\frac{1}{2}$$2  = $$\frac{1}{2}$$*$$\frac{1}{2}$$
=1 $$\frac{1×1}{2×2}$$
=1 $$\frac{1}{4}$$
= $$\frac{5}{4}$$

• Which of the fractions below are squares?

• $$\frac{4}{15}$$
Numerator is 4 and Denominator is 15 .
Both are not perfect Square numbers so it is not possible to write the given fraction as squares.

• $$\frac{8}{9}$$
Numerator is 8 and Denominator is 9.
Here 8 is not a perfect Square number and Denominator is a perfect square number. so it is not possible to write the given fraction as squares.

• $$\frac{16}{25}$$
In a given fraction, numerator is 16 and Denominator is 25 which were square numbers of 4 and 5.Hence we can Write as $$\frac{4}{5}$$2

• 2$$\frac{1}{4}$$
When the given 2$$\frac{1}{4}$$ is converted to improper fraction, we get $$\frac{9}{4}$$.
Here the Numerator is 9 and Denominator is 4
We can write the numerator as perfect square i.e. (3)2
Denominator 4 can be written as (2)2
Therefore, 2$$\frac{1}{4}$$ can be written as $$\frac{3}{2}$$2

• 4$$\frac{1}{9}$$
When the given 4$$\frac{1}{9}$$ is converted to improper fraction, we get $$\frac{1}{9}$$.
Here the Numerator is 1 and but the Denominator is 9 as a square of number (3)2
So, we cannot write the given fraction as Squares.

• $$\frac{8}{18}$$
Numerator is 8 and Denominator is 18.
Here numerator and Denominator are not perfect square numbers so there is no chance to write the given fraction as square of numbers.

Decimal squares

What is the square of 0.5?
We know that 52 = 25. How many decimal places would be there in the product 0.5 × 0.5?
Why?
0.5 = $$\frac{5}{10}$$, right?
Can you find out the square of 0.05?
0.052 =0.05×0.05
= 0.0025 Hence four decimal places are there.
0.0025 = $$\frac{25}{1000}$$

You have computed the squares of many natural numbers. Using that table, can you find out the square of 0.15?

• Find out the squares of these numbers.
0.152 = 0.15 x 0.15
= 0.0225

• 1.2
1.22 = 1.2 x 1.2
= 1.44

• 0.12
0.122 = 0.12 x 0.12
= 0.0144

• 0.013
0.0132 = 0.013 x 0.013
= 0.000169

• Which of the following numbers are squares?

• 2.5
We cannot write the number 2.5 as a square.

• 0.25
0.25 = 0.5 x 0.5.
Hence the number 0.25 is written as 0.52
• 0.0016
0.0016 = 0.04 x 0.04.
Hence the number 0.0016 is written as 0.042

• 14.4
We cannot write the number 14.4 as square.

• 1.44
1.44= 1.2×1.2
Hence the number 1.44 is written as 1.22

Square product Textbook Page No. 84

What is 52 × 42?
52 × 42 = 25 × 16 = ……..
There is an easier way:
52 × 42 = 5 × 5 × 4 × 4
= (5 × 4) × (5 × 4)
= 20 × 20
= 400
= (5 x 4) x (5 x 4)
= 20 x 20
= 400

Can you find out the products below like this, without pen and paper?

• 52 × 82
52 × 82 = 5 x 5 x 8 x 8
= (5 x 8) x (5 x 8)
= 40 x 40
= 1600

• 2.52 × 42
2.52 × 42 = 2.5 x 2.5 x 4 x 4
= (2.5 x 4) x (2.5 x 4)
= 10 x 10
= 100

• (1.5)2 × (0.2)2
(1.5)2 × (0.2)2 = 1.5 x 1.5 x 0.2 x 0.2
= (1.5 x 0.2) x (1.5 x 0.2)
= 0.3 x 0.3
= 0.009

What general rule did we use in all these?
The product of the squares of two numbers is equal to the square of their product.
How do we say this in algebra?
x2y2 = (xy)2, for any numbers x, y
let the three numbers be x, y, and z
x2y2z2 = (xyz)2, for any numbers x, y, and z.

Square factors

How do we write 30 as a product of prime numbers?
30 = 2 × 3 × 5
So how do we factorize 900?
900 = 302 = (2 × 3 × 5)2 = 22 × 32 × 52
Similarly, using the facts that 24 = 23 × 3 and 242 = 576, we get
576 = 242 = (23 × 3)2 = (23)2 × 32 = 26 × 32

Can you write each number below and its square as a product of prime powers?
• 35
35 = 5 x 7
Thus 35 can be written as product of prime powers 51 x 71

• 45
45 = 5 x 9
= 5 x 3 x 3
= 51 x 32
Thus, 45 can be written as 51 x 32

• 72
72 = 24 x 3
= 8 x 3 x 3
= 2 x 2 x 2 x 3 x 3
= 23  x 32
Thus, 72 can be written as 23  x 32

• 36
36 = 9 x 4
= 9 x 2 x 2
= 3 x 3 x 2 x 2
= 32 x 22
Thus, 36 can be written as 32 x 22

• 49
49 = 7 x 7
= 72
Thus, 49 can be written as 72

Did you note any peculiarity of the exponents of the factors of the squares?

Reverse computation

We have to draw a square; and its area must be 9 square centimetres.
How do we do it?
The area of a square is the square of the side.

So if the area is to be 9 square centimetres, what should be the side?
To draw a square of area 169 square centimetres, what should be the length of a side?
For that, we must find out which number squared gives 169. Looking up our table of squares, we find 132 = 169. So we must draw a square of side 13 centimetres.

Here, given a number we found out which number it is the square of. This operation is called extracting the square root.
That is, instead of saying the square of 13 is 169, we can say in reverse that the square root of 169 is 13.
Just as we write
132 = 169
as shorthand for statement “the square of 13 is 169”, we write the statement “the square root of 169 is 13” in shorthand form as
$$\sqrt{169}$$ = 13
(the extraction of square root is indicated by the symbol )
Similarly, the fact that the square of 5 is 25 can also be stated, the square root of 25 is 5. In short hand form,
52 = 25
$$\sqrt{25}$$ = 5
In general
For numbers x and y, if x2 = y, then $$\sqrt{y}$$ = x

Now find out the square root of these numbers:

• 100
10 = 100
$$\sqrt{100}$$ = 10
Therefore, square root of 100 is 10

• 256
162 = 256
$$\sqrt{256}$$ = 16
Thus, the square root of 256 is 16.

• $$\frac{1}{4}$$
$$\frac{1}{4}$$ = $$\frac{1}{2}$$2
Thus, the Square root of $$\frac{1}{4}$$ = $$\frac{1}{2}$$

• $$\frac{16}{25}$$
$$\frac{16}{25}$$ = $$\frac{4}{5}$$2
Thus, the square root of $$\frac{16}{25}$$ is $$\frac{4}{5}$$.

• 1.44
1.44 = (1.2)2
Thus, the square root of 1.44 is 1.2

• 0.01
0.01 = (0.1)2
Thus, the square root of 0.01 is 0.1

Square root factors Textbook Page No. 87

How do we find the square root of 1225?
Since a product of squares is the square of the product, we need only write 1225 as a product of squares.
First factorize 1225 into primes:
1225 = 52 × 72
And we can write
52 × 72 = (5 × 7)2 = 352
So, 1225 = 352
From this, we get $$\sqrt{1225}$$ = 35
Let’s take another example. What is the $$\sqrt{3969}$$ ?
As before, we first factorize 3969 into primes.
3969 = 32 × 32 × 72
= (3 × 3 × 7)2
From this, we get $$\sqrt{3969}$$ = 3 × 3 × 7 = 63

Now compute the square roots of these.
• 256
Given number is 256
Firstly factorizing it into primes we have 256 = 2 x 128
= 2 x 2 x 64
= 2 x 2 x 2 x 32
= 2 x 2 x 2 x 2 x 16
= 2 x 2 x 2 x 2 x 2 x 8
= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
= (2 x 2) x (2 x 2)  x (2 x 2) x (2 x 2)
= 4 x 4 x 4 x 4
= 16 x 16
= (16)2
From this, we get $$\sqrt{256}$$ = 16

• 2025
Given number is 2025
Writing the factorization of 2025 we have
= 3 x 3 x 3 x 3 x 5 x 5
= 9 x 9 x 5 x 5
= (9 x 5)2
= (45)2
Therefore, the square root of $$\sqrt{2025}$$ is 45

• 441
Given number is 441
Writing the factorization of 441 we have
= 3 x 3 x 7 x 7
= 32 x 72
= (3 x 7)2
= (21)2

Therefore, the square root of 441 is 21.

• 921
Writing the factorization of 921 we have 3 x 307
Thus, 921 can’t be written as perfect square and the square root of 921 isn’t a natural number.

• 1089
Given number is 1089
Writing the factorization of it we have 1089 = 3 x 3 x 11 x 11
= 32 x 112
= (3 x 11)2
= (33)2
Therefore, square root of 1089 is 33.

• 15625
Given number is 15625
Writing the factorization of it we have 15625 = 5 x 5 x 5 x 5 x 5 x 5
= 52 x 52 x 52
= (5 x 5 x 5)2
= (125)2
Therefore, the square root of 15625 is 125

• 1936
Given number is 1936
Writing the factorization of 1936 we have 2 x 2 x 2 x 2 x 11 x 11
= (22 x 22 x 112)
= (2 x 2 x 11)2
= 442
Therefore, the square root of 1936 is 44.

• 3025
Given number is 3025
Writing the factorization of 3025 we have 5 x 5 x 11 x 11
= 52 x 112
= (5 x 11)2
= 552
Therefore, the square root of 3025 is 55.

• 12544
Given number is 12544
Writing the factorization of 12544 we have 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 7
= 28 x 72
= (112)2
Therefore, Square Root of 12544 is 112.

Let’s do it!

Question 1.
The area of a square plot is 1024 square meters. What is the length of its sides?
Given area of Square plot = 1024 Square Meters
Imagine ‘S’ is side of a Square. As we know Area of Square = Side x Side
1024 = Side x Side
1024 = Side2
Side = √1024
Side = 2 5
Hence Length of Square Plot = 32 meters

Question 2.
In a hall, 625 chairs are arranged in rows and columns, with the number of rows equal to the number of columns. The chairs in one row and one column are removed. How many chairs remain?
Total number of chairs = 625
Number of Rows and Columns present are Equal i.e. Rows =  Columns , Let it be y.
Rows .Columns=625
y. y =625
y 2= 625
y = √625
y = √(25)2
y = 25
so Number of Rows and Number of Columns =25
When 1 Row and 1 Column is removed as below
Number of Rows =25-1=24
Number of Rows =25-1=24
When the 24 Rows and 24 Columns are arranged = 24 x 24
= 576

Question 3.
The sum of a certain number of consecutive odd numbers, starting with 1, is 5184. How many odd numbers are added?
We know the formula of sum of arithmetic series Sn = n/2[2a+(n-1)d]
The arithmetic series 1, 3, 5, 7, 9, 11, 13 ……
Here the first term a = 1
Number of odd numbers to be added = n
Common difference d = 2
Substituting in the formula above we have
5184 = n/2[2×1+(n-1)2]
5184 = n/2[2+2n-2]
5184 = n/2[2n]
5184 = n2
n = 72
Thus, the number of odd numbers to be added is 72.

Question 4.
The sum of two consecutive natural numbers and the square of the first is 5329. What are the numbers?
Suppose x and x+1 are two Consecutive natural numbers
As per the Given data, Square of First number x2  = 5329
When they are added it gives as  x+(x+1) +x2 = 5329
x + x+1+x2 = 5329
x + x+1+x = 5329
2x +1+x = 5329
(x+1)= 5329
(x+1) = √5329
x+1 = √(73)2
x+1 = 73
x = 73-1
x= 72
Therefore, the numbers are 72 and 73.

## Kerala State Syllabus 7th Standard Maths Solutions Chapter 9 Ratio

### Ratio Text Book Questions and Answers

Same shape Textbook Page No. 116

In both the rectangles below, the length is 1 centimetre more than the breadth.

But not only the length of these are different, they look different also. In the larger rectangle, the sides look almost the same. Draw a rectangle of length 50 centimetres and breadth 49 centimetres in a larger sheet of paper. It looks almost a square, doesn’t it?
In the first rectangle above, length is double the breadth. Now look at this rectangle.

Again the length is double the breadth. Even though it is larger than the first, they have the same shape, right?
As the length is double the breadth in the second rectangle. The first and the second rectangle will have the same shape.

Changing scale

Look at this photo:

The shorter side is 2 centimetres and the longer side, 3 centimetres; that is, the longer side is 1$$\frac{1}{2}$$ times shorter side.
Suppose we make the shorter side 3 centimetres and longer side 4.5 centimetres.

Still, the longer side is 1$$\frac{1}{2}$$ times the shorter side.
Now suppose we change the shorter side to 3 centimetres and increase the longer side also by 1 centimetre, making it 4 centimeters.

Does the picture look right?
Yes, the picture looks right.
Suppose if we change the shorter side to 3 centimetres and the longer side to 4 centimetres. The longer side is 1$$\frac{1}{3}$$ times the shorter side.

TV Math

The sizes of TV sets are usually given as 14 inch, 17 inch, 20 inch and so on. What does it mean?
The TV screen is a rectangle: and these are lengths of the diagonals of the screen.
Does it determine the size of the screen? Rectangles of different width and lengths can have the same diagonal:

In the modern TV sets, the ratio of length to width is 16 : 9. In the earlier days, it was 4 : 3.
See their difference in two TV screens of the same diagonal length.

Flags Textbook Page No. 119

When we draw our National Flag, not only should the colours be right, the ratio of width to length should also be correct. This ratio is 2 : 3.
That is, in drawing our flag, if the length is taken as 3 centimetres, the width should be 2 centimetres.

This ratio is different for flags of other countries. For example, in the flag of Australia, this ratio is 1 : 2.

And in the flag of Germany, it is 3 : 5.

Without fractions Textbook Page No. 120

When quantities like length are measured using a definite unit, we may not always get counting numbers; and it is this fact which led to the idea of fractions.

In comparing the sizes of two quantities, one question is whether both can be given as counting numbers, using a suitably small unit of measurement. It is this question that leads to the idea of ratio.

For example, suppose the length of two objects are found as $$\frac{2}{5}$$ and $$\frac{3}{5}$$, when measured using a string. If $$\frac{1}{5}$$ of the string is taken as the unit, we can say the length of the first is 2 and that of the second is 3. This is the meaning of
saying the ratio of the lengths is 2 : 3.
Suppose the length of two objects are $$\frac{1}{3}$$ and $$\frac{1}{5}$$ of the string.
To get both lengths as counting numbers, what fraction of the string can be taken as a unit of measurement?

Circle relations

Look at the pieces of circles below:

The smaller piece is $$\frac{1}{4}$$ of the circle and the larger piece is $$\frac{1}{2}$$ of the circle.
So, the larger piece is twice the size of the smaller. That is, the ratio of the sizes of small and large pieces is 1 : 2.
Now look at these pieces:

What is the ratio of their sizes?
Let’s measure each using $$\frac{1}{4}$$ of circles.
The smaller figure has two such pieces. What about the larger?
Let’s measure each using $$\frac{1}{4}$$ of circles.
The larger figure has three such pieces.

So what is the ratio of the sizes of these two?
Let’s measure each using $$\frac{1}{4}$$ of circles.
The smaller piece has two such pieces and the larger piece has three such pieces.
2×$$\frac{1}{4}$$ : 3×$$\frac{1}{4}$$
$$\frac{2}{4}$$ : $$\frac{3}{4}$$
Therefore, the ratio will be 2 : 3

Moving Ratio Textbook Page No. 122
Have you taken apart toy cars or old clocks? There are many toothed wheels in such mechanisms. See this picture:

It is a part of a machine. In the two whole wheels we see, the smaller has 13 teeth and the larger has 21. So during the time that the smaller circle makes 21 revolutions, the larger one would have made only 13 revolutions.
The speed of rotation of machines is controlled by arranging such wheels with different number of teeth.

Sand and Cement

In construction of a building, sand and cement are used in definite ratios, but the ratios are different depending on the purpose. When one bowl of cement and five bowls of sand are mixed, the ratio of cement to sand is 1 : 5

When one sack of cement is mixed with five sacks of sand, the ratio is the same. But to set a brick wall, this much cement may not be needed. In this case, the ratio may be 1 : 10 or 1 : 12.

Ratio of Parts Textbook Page No. 124

We can use ratio to compare parts of a whole also. For example, in the picture below, the lighter part is $$\frac{3}{8}$$ of the circle, and the darker part is $$\frac{5}{8}$$ of the circle.

These two parts together make up the whole circle. The ratio of the sizes of these parts is 3 : 5.

So here, the ratio 3 : 5 indicate the two fractions $$\frac{3}{8}$$ and $$\frac{5}{8}$$.

Generally in such instances, a ratio of the numbers indicates fractions of equal denominators adding up to 1.

Meaning of ratio Textbook Page No. 125

If we know only the ratio of two quantities, we can’t say exactly how much they are; but they can be compared in several ways.
For examples, suppose the volumes of two pots are said to be in the ratio 2 : 3. We can interpret this in the following ways:

• To fill the smaller pot, we need $$\frac{2}{3}$$ of the larger pot.
• To fill the bigger pot, we need $$\frac{3}{2}$$ = 1$$\frac{1}{2}$$ times the smaller pot.
• Whether we take $$\frac{1}{2}$$ of the water in the smaller pot, or $$\frac{1}{3}$$ of the water in the larger pot, we get the same amount of water.
• If we fill both pots and pour them into a large pot, $$\frac{2}{5}$$ of the total amount of water
is from the smaller pot and $$\frac{3}{5}$$ from the larger.

If the length of two pieces of rope are said to be in the ratio 3 : 5, what all interpretations like this can you make?
If the length of two pieces of rope is 3 : 5.Then the following interpretations can be made.
The smaller rope is $$\frac{3}{5}$$ of the length of the larger rope.
The smaller rope will be $$\frac{3}{8}$$ of the total rope length.

Three measures

Look at this triangle:

In it, the longest side is double the shortest. The side of medium length is one and a half times the shortest.
Saying this with ratios, the lengths of the shortest and the longest sides are in the ratio 1 : 2 and those of the shortest and the medium are in the ratio 2 : 3.
What is the ratio of the lengths of the medium side and the longest side?
We can say this in another way: if we use a string of length 1.5 centimetres, the shortest side is 2, the medium side is 3 and the longest side is 4.
The can be shortened by saying the sides are in the ratio 2 : 3 : 4.

Triangle Math Textbook Page No. 127

How many triangles are there with ratio of sides 2 : 3 : 4?
The lengths of sides can be 2 cm, 3 cm, 4cm.

Or they can be 1 cm, 1.5 cm, 2 cm

We can take metres instead of centimetres.
Thus we have several such triangles.
In all these, what fraction of the perimeter is the shortest side?
And the side of medium length?
The longest side?
The perimeter of a triangle is 80 centimetres and its sides are in the ratio 5 : 7 : 8. Can you compute the actual lengths of sides?
Let the ratio be x.Then the sides will be 5x, 7x and 8x.
Given that perimeter is 80 cm.
5x + 7x + 8x = 80
20x = 80
x = 4
Since the value of x is 4, the lengths will be 5×4=20cm, 7×4=28cm and 8×4=32cm.

What if the perimeter is 1 metre?
1 metre equals 100 centimetres.
Let the ratio be x.Then the sides will be 5x, 7x and 8x.
Given that perimeter is 100 cm.
5x + 7x + 8x = 100
20x = 100
x = 5
Since the value of x is 5, the lengths will be 5×5=25cm, 7×5=35cm and 8×5=40cm.

Length and width Textbook Page No. 116

Look at these rectangles:

Is there any common relation between the lengths and widths of all these?
In all these, the length is twice the width, right? (We can also say width is half the length)
In the language of mathematics, we state this fact like this:
In all these rectangles, the width and length are in the ratio one to two.
In writing, we shorten the phrase “one to two” as 1 : 2; that is
In all these rectangles, the width and length are in the ratio 1 : 2.

In the rectangle of width 1 centimetre and length 2 centimetres, the length is twice the width. In a rectangle of width 1 metre and length 2 metres also, we have the same relation. So, in both these rectangles, width and length are in the ratio one to two (1 : 2).
We can state this in reverse: in all these rectangles, length and width are in the ratio two to one (2 : 1).
What is the width to length ratio of the rectangle below?

The length of the given rectangle is 6cm and width is 2cm.
Therefore, the width to length ratio for the above rectangle will be 1 : 3.

The length of the given rectangle is 4.5cm and width is 1.5cm.
Therefore, the width to length ratio for the above rectangle will be 1 : 3.

In both, the length is 3 times the width. So what is the ratio of the width to the length?
In the above two rectangles, the length is thrice the width.
The ratio of the width to the length is 1 : 3.

How do we say this as a ratio?
The width and length are in the ratio one to three.

1 metre means loo centimetres. So in such a rectangle, the ratio of the width to the length is 1: 50.
Now look at these rectangles:

In both, the length is one and a half times the width.
How do we say this as a ratio?
We can say one to one and a half; but usually we avoid fractions when we state ratios.
Suppose we take the width as 2 centimetres.
What is 1$$\frac{1}{2}$$ times 2?

So, we say that in rectangles of this type, the ratio of width to length is two to three and write 2 : 3.
Can’t we say here that the ratio is 4 : 6 ?
Nothing wrong in it; but usually ratios are stated using the least possible counting numbers.
So how do we state using ratios, the fact that the length of a rectangle is two and a half times the width?

If the width is 1 centimetre, then the length is 2$$\frac{1}{2}$$ centimetres.
What if the width is 2 centimetres?
Length would be 5 centimetres.
So, we say that width and length are in the ratio 2 to 5.

What if the length is one and a quarter times the width?
If the width is 1 centimetre, the length is 1$$\frac{1}{4}$$ centimetre.
If the width is 2 centimetres then the length is 2$$\frac{1}{2}$$ centimetres.
Still we haven’t got rid of fractions.
Now if width is 4 centimetres, what would be the length?

So, in all such rectangles, the width and length are in the ratio 4 : 5.
Do you notice another thing in all these?
If we stretch or shrink both width and length by the same factor, the ratio is not changed. For example, look at this table.

In all these, the length is 3 times the width; or the reverse, the length is $$\frac{1}{3}$$ of the width.
In terms of ratio, we say width and length are in the ratio 1 : 3 or length and width are in the ratio 3 : 1.
In terms of ratio, we say width and length are in the ratio 1: 3 or length and width are in the ratio 3: 1.

• For all rectangles of dimensions given below, state the ratio of width to length, using the least possible counting numbers:

• width 8 centimetres, length 10 centimetres
The simplest form for the given numbers 8:10 is 4 : 5
Therefore, the ratio of width to length will be 4 : 5

• width 8 metres length 12 metres
The simplest form for the given numbers 8:12 is 4 : 3
Therefore, the ratio of width to length will be 2 : 3

• width 20 centimetres length 1 metre
The given numbers cannot be written in simplest form.
Therefore, the ratio of width to length will be 20 : 1

• width 40 centimetres length 1 metre
The given numbers cannot be written in simplest form.
Therefore, the ratio of width to length will be 40 : 1

• width 1.5 centimetres length 2 centimetres
The simplest form for the given numbers 1.5:2 is 3 : 4
Therefore, the ratio of width to length will be 3 : 4

• In the table below two of the width, length and their ratio of some rectangles are given. Calculate the third and fill up the table.

• What does it mean to say that the width to length ratio of a rectangle is 1: 1 ? What sort of a rectangle is it?
The width to length ratio of a rectangle is 1 : 1.
Here the width of the rectangle and length of the rectangle is same.
If the width and the length of rectangle is same, then it is a square.

Other quantities Textbook Page No. 120

There are two ropes, the shorter $$\frac{1}{3}$$ metre long and the longer, $$\frac{1}{2}$$ metre long. What is the ratio of their lengths?

We can do this in different ways.
We can check how much times $$\frac{1}{3}$$ is $$\frac{1}{2}$$.
$$\frac{1}{2}$$ ÷ $$\frac{1}{2}$$ = $$\frac{3}{2}$$
So, the length of the longer rope is $$\frac{3}{2}$$ times that of the shorter.
That is, 1$$\frac{1}{2}$$ times.
If the length of the shorter rope is taken as 1, the length of the longer is 1$$\frac{1}{2}$$; if 2, then 3.
So the length of the shorter and longer rope are in the ratio 2 : 3.

We can also think in another manner. As in the case of the width and length of rectangles, we can imagine the length to be stretched by the same factor; and the ratio won’t change.
Suppose we double the length of each piece of rope.
Then the length of the shorter is $$\frac{2}{3}$$ metres and that of the longer 1 metre. This doesn’t remove fractions.

By what factor should we stretch to get rid of fractions? How about 6 ?
6 times $$\frac{1}{3}$$ is 2.
6 times $$\frac{1}{2}$$ is 3.
The shorter is now 2 metres and the longer is 3 metres. So the ratio is 2 : 3.
There is yet another way; we can write
$$\frac{1}{3}$$ = $$\frac{2}{6}$$ $$\frac{1}{2}$$ = $$\frac{3}{6}$$
That is, we can think of the shorter rope as made up of 2 pieces of $$\frac{1}{6}$$ metres each and the longer one as 3 of the same $$\frac{1}{6}$$ metres. In this way also, we can calculate the ratio as 2 to 3.
Now look at this problem : to fill a can we need only half the water in a bottle.

To fill a larger can, we need three quarters of the bottle. What is the ratio of the volume of the smaller can to the larger can ?
Here, we can write
$$\frac{1}{2}$$ = $$\frac{2}{4}$$
So, if we pour $$\frac{1}{4}$$ of the bottle 2 times, we can fill the smaller can; to fill the larger can, $$\frac{1}{4}$$ of the bottle should be poured 3 times. So the volumes of the smaller and larger cans are in the ratio 2 : 3.

Another problem: Raju has 200 rupees with him and Rahim has 300. What is the ratio of the money Raju and Rahim have?
If we imagine both to have the amounts in hundred rupee notes, then Raju has 2 notes and Rahim, 3 notes. So the ratio is 2 : 3.

Let’s change the problem slightly and suppose Raju has 250 rupees and Rahim, 350 rupees.
If we think of the amounts in terms of 50 rupee notes, then Raju has 5 and Rahim has 7.
The ratio is 5 : 7.
What if the amounts are 225 and 325 rupees?

Imagine each amount as packets of 25 rupees. Raju has 225 ÷ 25 = 9 packets and Rahim has 325 ÷ 25 = 13 packets. The ratio is 9 : 13.
Let’s look at one more problem: in a class, there are 25 girls and 20 boys. What is the ratio of the number of girls to the number of boys?
If we split the girls and boys separately into groups of 5, there will be 5 groups of girls and 4 groups of boys. So the ratio is 5 to 4.

In this way, calculate the required ratios and write them using the least possible counting numbers, in these problems:

• Of two pencils, the shorter is of length 6 centimetres and the longer, 9 centimetres. What is the ratio of the lengths of the shorter and the longer pencils?
Given that the length of shorter pencil is 6 centimetres and the longer is 9 centimetres.
The simplest form of 6 and 9 is 2 and 3.
In terms of ratio, the ratio of the lengths of the shorter and the longer pencils 2 : 3.

• In a school, there are 120 boys and 140 girls. What is the ratio of the number of boys to the number of girls?
Given that there are 120 boys and 140 girls in a school.
The simplest form of 120 and 140 is 6 and 7.
In terms of ratio, it will be 6 : 7

• 96 women and 144 men attended a meeting. Calculate the ratio of the number of women to the number of men.
Given that 96 women and 144 men attended a meeting.
The simplest form of 96 and 144 is  and 22 and 3.
In terms of ratio, it will be 2 : 3

• When the sides of a rectangle were measured using a string, the width was $$\frac{1}{4}$$ of the string and the length $$\frac{1}{3}$$ of the siring. What is the ratio of the width to the length’?
Given that the width of rectangle is $$\frac{1}{4}$$ of the string and the length is $$\frac{1}{3}$$ of the siring.
The ratio of the width to the length will be 3 : 4

• To fill a larger bottle, 3$$\frac{1}{2}$$ glasses of water are needed and to fill a smaller bottle, 2$$\frac{1}{4}$$ of glasses are needed. What is the ratio of the volumes of the large and small bottles?
Given that to fill a larger bottle 3$$\frac{1}{2}$$ glasses of water are needed and to fill a smaller bottle, 2$$\frac{1}{4}$$ of glasses are needed.
3$$\frac{1}{2}$$ equals $$\frac{7}{2}$$ and 2$$\frac{1}{4}$$ equals $$\frac{9}{4}$$.
The ratio of the volumes of the large and small bottles will be 14 : 9

Ratio of mixtures Textbook Page No. 123

To make idlis , Ammu’s mother grinds two cups of rice and one cup of urad dal. When she expected some guests the next day, she took four cups of rice. How many cups of urad should she take?
To have the same consistency and taste, the amount of urad must be halfthe amount of rice.
So for four cups of rice, there should be two cups of urad.
We can say that the quantities, rice and urad must be in the ratio 2 :1
Now another problem on mixing: to paint the walls of Abu’s home, first 25 litres of green and 20 litres of white were mixed.

When this was not enough, 15 litres of green was taken. How much white should be added to this?
To get the same final colour, the ratio of green and white should not change.
In what ratio was green and white mixed first?
That is, for 5 litres of green, we should take 4 litres of white.
To maintain the same ratio, for 15 litres of green, how many litres of white should we take?
How many times 5 is 15?
So 3 times 4 litres of white should be mixed.
That is 12 litres.
To get the same shade of green, how many litres of green should be mixed with 16 litres of white?

Now try these problems:

• For 6 cups of rice, 2 cups of urad should be taken to make dosas. How many cups of urad should taken with 9 cups of rice?
Given that 6 cups of rice, 2 cups of urad should be taken to make dosas.
In terms of ratio, the ratio of the rice to urad will be 3 : 1.
Here, the number of rice cups is thrice the number of urad cups.
Therefore, to maintain the same ratio, for 9 cups of rice, 3 cups of urad should be taken.

• To set the walls of Nizar’s house, cement and sand were used in a ratio 1:5. He bought 45 sacks of cement. How many sacks of sand should he buy?
Given that to set the walls of Nizar’s house, cement and sand were used in a ratio 1:5.
Here, 1 resembles the number of units for cement sacks and 5 resembles the number of unit for sand sacks.
The number of sand sacks required is 5 times the number of cement sacks.
If there are 45 sacks of cement, then the number of sand sacks required will be 45×5=225.

• To paint a house, 24 litres of paint was mixed with 3 litres of turpentine. How many litres of turpentine should be mixed with 32 litres of paint?
Given that to paint a house, 24 litres of paint was mixed with 3 litres of turpentine.
In terms of ratio, the ratio of the paint to turpentine will be 8 : 1.
Let x be the number of litres of turpentine that should be mixed with 32 litres of paint.
$$\frac{8}{1}$$ = $$\frac{32}{x}$$
8x = 32
x = $$\frac{32}{8}$$
x = 4
Therefore, 4 litres of turpentine should be mixed with 32 litres of paint.

• In the first ward of a panchayat, the male to female ratio is 10 : 11. There are 3311 women. How many men are there? What is the total population in that panchayat?
Given that In the first ward of a panchayat, the male to female ratio is 10 : 11.
Let the number of men be m and there are 3311 women.
$$\frac{10}{11}$$ = $$\frac{m}{3311}$$
11 × m  = 10 × 3311
m = $$\frac{10 × 3311}{11}$$
m = 3010
Therefore, the number of mens will be 3010.
The total population in that panchayat will be the sum of number of mens and womens, which is 3010+3311=6321

• In a school, the number of female and male teachers are in the ratio 5 : 1. There are 6 male teachers. How many female teachers are there?
Given that, In a school, the number of female and male teachers are in the ratio 5 : 1.
Here, the number of female teachers is five times the number of male teachers.
If there are 6 male teachers.Then the female teachers will be 5×6=30.
Thus, there are 30 female teachers.

• Ali and Ajayan set up a shop together. Ali invested 5000 rupees and Ajayan, 3000 rupees. They divided the profit for a month in the ratio of their investments and Ali got 2000 rupees. How much did Ajayan get? What is the total profit?
Given that, Ali invested 5000 rupees and Ajayan, 3000 rupees.
In terms of ratio, the ratio of Ali to Ajayan will be 5 : 3
Ali got 2000 rupees.Let m be the amount Ajayan should get.
$$\frac{5}{3}$$ = $$\frac{2000}{m}$$
5×m = 2000×3
m = $$\frac{2000×3}{5}$$
m = 1200
Ajayan will get 1200 rupees.
The total profit will be the sum of maount got by Ali and Ajayan, which is 2000+1200=3200 rupees.

Division Problem Textbook Page No. 125

We have seen that to make idlis, rice and urad are to be taken in the ratio 2:1. In 9 cups of such a mixture of rice and urad, how many cups of rice are taken?
2 cups of rice and 1 cup of urad together make 3 cups of mixture.
Here we have 9 cups of mixture.
How many times 3 is 9 ?
To maintain the same ratio, both rice and urad must be taken 3 fold.
So 6 cups of rice and 3 cups of urad.

Another problem: In a co-operative society there are 600 members are male and 400 are female. An executive committee of 30 members is to be formed with the same male to female ratio as in the society. How many male and female members are to be there in the committee?
In the society, the male to female ratio is 3 : 2.
3 men and 2 women make 5 in all.
Here we need a total of 30.
How many times 5 to 30?
So there should be 3 × 6 = 18 men and 2 × 6 = 12 women in the committee.

One more problem: a rectangular piece of land is to be marked on the school ground for a vegetable garden. Hari and Mary started making a rectangle with a 24 metre long rope. Vimala Teacher said it would be nice, if the sides are in the ratio 3 : 5. What should be the length and width of the rectangle?
The length of the rope is 24 metres and this is the perimeter of the rectangle.
Given that, Hari and Mary started making a rectangle with a 24 metre long rope and their teacher suggested to make it in 3 : 5 ratio.
Let the ratio be x. Then the sides will be 3x and 5x.
Perimeter will be the total length of the rope which is 24 metre long.
3x + 5x = 24
8x = 24
x = $$\frac{24}{8}$$
x = 3
Therefore, the ength and width of the rectangle will be 9 metre and 15 metre.

If we take the length and width as 3 meters and 5 metres, what would be the perimeter?
If the length and width is 3 meters and 5 metres respectively.
Then the perimeter will be 3+5=8 metres.

How much of 16 is 24?
$$\frac{24}{16}$$ = $$\frac{3}{2}$$ = 1$$\frac{1}{2}$$
So width should be 1$$\frac{1}{2}$$ times 3 metres; that is
3 × 1$$\frac{1}{2}$$ = 4$$\frac{1}{2}$$ metres.
And length should be 1$$\frac{1}{2}$$ times 5 metres, that is,
5 × 1$$\frac{1}{2}$$ = 7$$\frac{1}{2}$$ metres

Now try these problems:

• Suhra and Sita started a business together. Suhra invested 40000 rupees and Sita, 30000 rupees. They made a profit of 7000 rupees which they divided in the ratio of their investments. How much did each get?
Suhra invested 40000 rupees and Sita, 30000 rupees in a business together.
In terms of ratio, the ratio of Suhra to Sita will be 4 : 3.
They made a profit of 7000 rupees.
Let the ratio be x, then the investment will be 4x and 3x.
4x + 3x = 7000
7x = 7000
x = 1000
Therefore, Suhra’s profit will be 4×1000=4000 rupees
Sita’s profit will be 3×1000=3000 rupees.

• John and Ramesh took up a job on contract. John worked 7 days and Ramesh, 6 days. They got 6500 rupees as wages which they divided in the ratio of their investments. How much did each get?
Given that John worked 7 days and Ramesh, 6 days.
In terms of ratio, the ratio for the number of days worked by John to Ramesh will be 7 : 6
Let the ratio be x, then the investment will be 7x and 6x.
7x + 6x = 6500
13x = 6500
x = $$\frac{6500}{13}$$
x = 500
Therefore, John will get 7×500 = 3500 rupees
Ramesh will get 6×500 = 3000 rupees

• John and Ramesh took up a job on contract. John worked 7 days and Ramesh, 6 days. They got 6500 rupees as wages which they divided in the ratio of the numbers of days each worked. How much did each get?
Given that John worked 7 days and Ramesh, 6 days.
In terms of ratio, the ratio for the number of days worked by John to Ramesh will be 7 : 6
Let the ratio be x, then the investment will be 7x and 6x.
7x + 6x = 6500
13x = 6500
x = $$\frac{6500}{13}$$
x = 500
Therefore, John will get 7×500 = 3500 rupees
Ramesh will get 6×500 = 3000 rupees

• Angles of a linear pair are in the ratio 4 : 5. What is the measure of each angle?
Angles of a linear pair are in the ratio 4 : 5.
Let the ratio be x,then it will be 4x and 5x.
4x + 5x = 90
9x = 90
x =10
4x = 4 × 10 = 40
5x = 5 × 10 = 50
Thus, the angles measurement will be 40 degrees and 50 degrees.

• Draw a line AB of length 9 centimetres. A point P is to be marked on it, such that the lengths of AP and PB are in the ratio 1 : 2. How far from A should P be marked? Compute this and mark the point.
Length of AB is 9 centimetres.
Point P is marked in 1 : 2 ratio.
Let the ratio be x, then it will be 1x and 2x, the total is 3x.
$$\frac{1x}{3x}$$ × 9 = 3 centimetres
$$\frac{2x}{3x}$$ × 9 = 6 centimetres

P should be marked 3 cm away from A.

• Draw a line 15 centimetres long. A point is to be marked on it, dividing the length in the ratio 2 : 3. Compute the distances and mark the point.
Length of AB is 15 centimetres.
Point P is marked in 2 : 3 ratio.
Let the ratio be x, then it will be 2x and 3x, the total is 5x.
$$\frac{2x}{5x}$$ × 15 = 6 centimetres
$$\frac{3x}{5x}$$ × 15 = 9 centimetres

P should be marked 6 cm away from A.

• Sita and Soby divided some money m the ratio 3 : 2 and Sita got 480 rupees. What is the total amount they divided?
Sita and Soby divided some money ‘m’ in the ratio 3 : 2 and Sita got 480 rupees.
Let the ratio be m, then the ratio of money Sita got will be 3m.
3m = 480
m = $$\frac{480}{3}$$
m = 160
Money Soby will get 2×160 = 320 rupees.

• In a right triangle, the two smaller angles are in the ratio 1 : 4. Compute these angles.
In a right triangle, the two smaller angles are in the ratio 1 : 4.
Right triangle means 90 degrees.
Let the ratio be x, total will be 1x + 4x = 5x.
$$\frac{1x}{5x}$$×90 = 18 degrees
$$\frac{4x}{5x}$$×90 = 72 degrees
Then the angles will be 18 degrees and 72 degrees.

• Draw a rectangle of perimeter 30 centimetres and lengths of sides in the ratio 1 : 2. Draw two more rectangles of the same perimeter, with lengths of sides in the ratio 2 : 3 and 3 : 7. Compute the areas of all three rectangles.
A rectangle of perimeter 30 centimetres is drawn with lengths of sides in the ratio 1 : 2
Let the ratio be x, the total will be 1x + 2x = 3x.
One of the side will be $$\frac{1x}{3x}$$×30 = 10 centimetres
The other side will be $$\frac{2x}{3x}$$×30 = 20 centimetres
Therefore, the rectangle with 10 centimetres and 20 centimetres is drawn as shown below.
Area of this rectangle will be 10×5=50 square centimetres.

A rectangle of perimeter 30 centimetres is drawn with lengths of sides in the ratio 2 : 3
Let the ratio be x, the total will be 2x + 3x = 5x.
One of the side will be $$\frac{2x}{5x}$$×30 = 12 centimetres
The other side will be $$\frac{3x}{5x}$$×30 = 18 centimetres
Therefore, the rectangle with 12 centimetres and 18 centimetres is drawn as shown below.
Area of this rectangle will be 9×6=54 square centimetres.

A rectangle of perimeter 30 centimetres is drawn with lengths of sides in the ratio 3 : 7
Let the ratio be x, the total will be 3x + 7x = 10x.
One of the side will be $$\frac{3x}{10x}$$×30 = 9 centimetres
The other side will be $$\frac{7x}{10x}$$×30 = 21 centimetres
Therefore, the rectangle with 9 centimetres and 21 centimetres is drawn as shown below.
Area of this rectangle will be 10.5×4.5=47.25 square centimetres.

## Kerala State Syllabus 6th Standard Maths Solutions Chapter 4 Volume

### Volume Text Book Questions and Answers

Large and small Textbook Page No. 57

Athira has collected many things and has arranged them into different lots.

Look at two things from the first lot.

Which is bigger?
How did you find out?
Now look at two things from the second lot:

How do we find out which is bigger?
To find out the bigger of two sticks, we need only measure their lengths.
We have to calculate their area, right?
Yes,

Explanation:
Yes, by calculating the area we can find which of the two envelopes is bigger.

Rectangle blocks
Look at two wooden blocks from Athira’s collection. Which is larger?

How did you decide?
Now look at these two.
Which is larger?

Let’s see how we can decide.
The block with more volume is larger,

Explanation:
We can decide about a larger wooden block by comparing their volumes. Volume = length X breadth X height.

Size of a rectangle block Textbook Page No. 59

Look at these rectangular blocks:

They are all made by stacking smaller blocks of the same size.
Which of them is the largest?
We need only count the little blocks in each, right?
Can you find how many little blocks make up each of large blocks below?
Is there a quick way to find the number of little blocks in each, without actually counting all?

Yes, we can find the number of little blocks which make up the large block. Yes, there is quick way to find the number of little blocks ; Count the number of blocks in length , breadth and height wise.
By the product of these values we can get the total number of small blocks.

This rectangular block contains 64 smaller blocks. If one small block is removed from each corner of the large block
above, how many would be left?
56 small blocks are left,

Explanation:
Total number of small blocks in rectangular block = 64, Number of corners in a rectangular block = 8, Number of small blocks left in the rectangular block = 64 – 8 = 56.

Which of these is the largest?
And the smallest?
Second block is larger than the first,

Explanation:
Volume of the first rectangular block = length X breadth X height = 3 X 3 X 3 = 27 cubic units,
Volume of the second rectangular block = length X breadth X height = 5 X 3 X 4 = 60 cubic units,
Therefore, second block is larger than the first.

Look at these blocks:

How many small blocks are there in each?
Do they have the same size?
To compare sizes by just counting, what kind of little blocks should be used in both?
Same shape and size,

Explanation:
Each block consists of 12 small blocks. Yes, the two blocks are of same size. The little blocks used should be of same shape and size.

Size as number Textbook Page No. 60

Look at this picture:

What is the area of the rectangle?
How many small squares of side 1 centimetre are in it?
4 X 3 = 12
The area of a square of side 1 centimetre is 1 square centimetre; the area of the whole rectanlge is 12 square centimetres.
Now look at the rectangular block:

It is made by stacking cubes of side 1 centimetre.

How many?
So, the size of this block is equal to 24 such cubes.
Size measured like this is called volume in mathematics.
We say that a cube of length, breadth and height 1 centimetre has a volume of 1 cubic centimetre.
24 such cubes make up the large block in the picture.
Its volume is 24 cubic centimetre.

All sides of the large cube shown above are painted. How many small cubes would have no paint at all?
One cube is left,

Explanation:
Only one cube is left with no paint on it which is in the middle of the large cube.

All blocks shown below are made up of cubes of side 1 centimetre. Calculate the volume of each:

64 cubic centimetres,12 cubic centimetres,64 cubic centimetres,36 cubic centimetres,63 cubic centimetres,12 cubic centimetres,

Explanation:

Total number of cubes = length X breadth X height = 4 X 4 X 4 = 64, if side of each small cube is 1 centimetre then Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 64 cubic centimetres.

Total number of cubes = length X breadth X height = 2 X 2 X 3 = 12, if side of each small cube is 1 centimetre then
Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 12 cubic centimetres.

Total number of cubes = length X breadth X height = 4 X 4 X 4 = 64, if side of each small cube is 1 centimetre then
Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 64 cubic centimetres.

Total number of cubes = length X breadth X height = 4 X 3 X 3 = 36, if side of each small cube is 1 centimetre then
Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 36 cubic centimetres.

Total number of cubes = length X breadth X height = 7  X 3 X 3 = 63, if side of each small cube is 1 centimetre then
Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 63 cubic centimetres.

Total number of cubes = length X breadth X height = 2 X 3 X 2 = 12, if side of each small cube is 1 centimetre then
Volume of each small cube = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres, Therefore volume of the figure = total number of cubes X volume of each small cube = 12 cubic centimetres.

Volume calculation Textbook Page No. 62

See this rectangular block:

How do we calculate its volume?
Volume is 15 cubic centimetre,

Explanation:
Volume of the rectangular block = length X breadth X height = 5 X 3 X 1 = 15 cubic centimetre.
Therefore, it’s volume is 15 cubic centimetre.

For that, we must find out how many cubes of side 1 centimetre we need to make it.

So, its volume is 15 cubic centimetres.

This can be made by stacking one over another, two blocks seen first:

So, to make it, how many cubes of side 1 centimetre do we need?

Thus the volume of this block is 30 cubic centimetres.

Like this, calculate the volume of each of the rectangular blocks shown below and write it beside each:

1) 56 cubic centimetres,
2) 54 cubic centimetres,
3) 125 cubic centimetres,
4) 100 cubic centimetres,

Explanation:

1) Length = 7 centimetres, Breadth = 4 centimetres, Height = 2 centimetres,
Volume of the rectangular block = length X breadth X height = 7 X 4 X 2 = 56 cubic centimetres.

2) Length = 6 centimetres, Breadth = 3 centimetres, Height = 3 centimetres
Volume of the rectangular block = length X breadth X height = 6 X 3 X 3 = 54 cubic centimetres.

3) Length = 5 centimetres, Breadth = 5 centimetres, Height = 5 centimetres,
Volume of the rectangular block = length X breadth X height = 5 X 5 X 5 = 125 cubic centimetres.

4) Length = 5 centimetres, Breadth = 4 centimetres, Height = 5 centimetres
Volume of the rectangular block = length X breadth X height = 5 X 4 X 5 = 100 cubic centimetres.

So, now, you know how to calculate the volume of a rectangular block, dont’ you?
The volume of a rectangular block is the product of its length, breadth and height.

Question 1.
The length, breadth and height of a brick are 21 centimetres, 15 centimetres and 7 centimetres. What is its volume?
Volume is 2205 cubic centimetres,

Explanation:
Volume of the rectangular block = length X breadth X height = 21 X 15 X 7 = 2,205 cubic centimetres.

Question 2.
A rectangular cube of iron is of side 8 centimetres. What is its volume? 1 cubic centimetre of iron weighs 8 grams. What is the weight of the large cube?
Weight of large cube is 4096 grams,

Explanation:
Volume of a rectangular cube = side X side X side = 8 X 8 X 8 = 512 cubic centimetres, Weight of 1 cubic centimetre of iron = 8 grams, Weight of the large cube = 512 cubic centimetres X 8 grams = 4,096 grams, Therefore, weight of the large cube is 4,096 grams or 4.096 kilo grams.

Volume and length Textbook Page No. 65

A wooden block of length 8 centimetres and breadth 4 centimetres has a volume of 180 cubic centimetres. What is its height?
Volume is the product of length, breadth and height.
So in this problem, the product of 9 and 4 multiplied by the height is 180.
That is, 36 multiplied by the height gives 180.
So to find out the height, we need only divide 180 by 36.
The table shows measurement of some rectangular blocks. Calculate the missing measures.

Explanation:
1) Length = 3 centimetres, Breadth = 8 centimetres, Height = 7 centimetres,
Volume of the rectangular block = length X breadth X height = 3 X 8 X 7 = 168 cubic centimetres.
2)Length = 6 centimetres, Breadth = 4 centimetres, Height = 5 centimetres,
Volume of the rectangular block = length X breadth X height = 6 X 4 X 5 = 120 cubic centimetres.
3)Length = 6 centimetres, Breadth = 4 centimetres, Volume of the rectangular block = length X breadth X height = 6 X 4 X height = 48 cubic centimetres , Height = $$\frac{48}{24}$$ = 2 centimetres.
4)Length = 8 centimetres, Height = 2 centimetres, Volume of the rectangular block = length X breadth X height = 8 X breadth X 2 = 48 cubic centimetres, Breadth = $$\frac{48}{16}$$ = 3 centimetres.
5) Breadth = 2 centimetres, Height = 2 centimetres, Volume of the rectangular block = length X breadth X height = length X 2 X 2 = 48 cubic centimetres, Length = $$\frac{48}{4}$$ = 12 centimetres.
6) Breadth = 2 centimetres, Height = 4 centimetres, Volume of the rectangular block = length X breadth X height = length X 2 X 4 = 80 cubic centimetres, Length = $$\frac{80}{8}$$ = 10 centimetres.
7)Length = 14 centimetres, Height = 5 centimetres, Volume of the rectangular block = length X breadth X height = 14 X breadth X 5 = 210 cubic centimetres, Breadth = $$\frac{210}{70}$$ = 3 centimetres.

Area and volume

What is the area of a rectangle of length 8 centimetres and breadth 2 centimetres?
What about the volume of a rectangular block of length 8 centimetres, breadth 2 centimetres and height 1 centimetre?
Area is 16 square centimetres, Volume is 16 cubic centimetres,

Explanation:
Length= 8 centimetres, Breadth= 2 centimetres, Height= 1 centimetre. Area of rectangle = length X breadth = 8 X 2 = 16 square centimetres, Volume of rectangle = length X breadth X height = 8 X 2 X 1 = 16 cubic centimetres.

New shapes Textbook Page No. 66

We can make shapes other than rectangular block, by stacking cubes. For example, see this:

It is made by stacking cubes of side I centimetre. Can you calculate its volume?
How many cubes are there at the very bottom?
And in the step just above it?
Thus we can count the number of cubes in each step.
How many cubes in all?
What is the volume of the stairs?
Volume of the stairs is 216 cubic centimetres,

Explanation:
Yes, we can calculate the volume of the given figure. There are total of 81 cubes in the bottom line because there are 9 cubes vertically and 9 cubes horizontally arranged product of these gives the number of cubes. There are total of 63 cubes in the bottom line because there are 7 cubes vertically and 9 cubes horizontally arranged product of these gives the number of cubes.
There are total of 216 cubes in the given figure obtained by adding the cubes of each line.
Volume of the stairs = total number of cubes ( since 1 cube is of 1 cubic centimetre in volume ) = 81 + 63 + 45 + 27 = 216 cubic centimetre.

Now look at this figure:

It is made by stacking square blocks. The bottom block is of side 9 centimetres. As we move up, the sides decrease by 2 centimetres at each step.

What is the volume of a rectangular block of length 4 centimetre, breadth 3 centimetre and
height 1 centimetre? If the length, breadth and height are doubled, what happens to the volume?
Volume of rectangular block is 12 cubic centimetres. If the length, breadth and height of a rectangular block are doubled then the volume increases by 8 times.

Explanation:
Length = 4 centimetre, Breadth = 3 centimetre, Height = 1 centimetre,
Volume of rectangular block = length X breadth X height = 4 X 3 X 1 = 12 cubic centimetres.

If the length, breadth and height are doubled then Length = 8 centimetre, Breadth = 6 centimetre
Height = 2 centimetre,Volume of rectangular block = length X breadth X height = 8 X 6 X 2 = 96 cubic centimetres.

Therefore, if the length, breadth and height of a rectangular block are doubled then the volume increases by 8 times.

All blocks are of height 1 centimetre. What is the volume of this tower?
Just calculate the volume of each square block and add. Try it!
Volume of the tower is 165 cubic centimetres,

Explanation:
Length = 9 centimetre, Breadth = 9 centimetre, Height = 1 centimetre, Volume of first square block = length X breadth X height = 9 X 9 X 1 = 81 cubic centimetres.
Length = 7 centimetre, Breadth = 7 centimetre , Height = 1 centimetre, Volume of second square block = length X breadth X height = 7 X 7 X 1 = 49 cubic centimetres.
Length = 5 centimetre, Breadth = 5 centimetre, Height = 1 centimetre, Volume of third square block = length X breadth X height = 5 X 5 X 1 = 25 cubic centimetres.
Length = 3 centimetre, Breadth = 3 centimetre, Height = 1 centimetre, Volume of fourth square block = length X breadth X height = 3 X 3 X 1 = 9 cubic centimetres.
Length = 1 centimetre, Breadth = 1 centimetre,  Height = 1 centimetre, Volume of fifth square block = length X breadth X height = 1 X 1 X 1 = 1 cubic centimetres. Volume of the tower = 81 + 49 + 25 + 9 + 1 = 165 cubic centimetres.

Calculate the volumes of the figures shown below. All lengths are in centimetres.

1) 416 cubic centimetres, 2) 448 cubic centimetres, 3) 324 cubic centimetres,

Explanation:

1) Volume of the given figure = Volume of vertical cuboid + Volume of horizontal cuboid + Volume of horizontal cuboid + Volume of cube, Length = 20 centimetre, Breadth = 4 centimetre, Height = 2 centimetre, Volume of vertical cuboid = length X breadth X height = 20 X 4 X 2 = 160 cubic centimetres, Length = 12 centimetre, Breadth = 4 centimetre, Height = 2 centimetre, Volume of horizontal cuboid = length X breadth X height = 12 X 4 X 2 = 96 cubic centimetres, Length of side of cube = 4 centimetres, Volume of cube = side X side X side = 4 X 4 X 4 = 64 cubic centimetres, Volume of the given figure = 160 cubic centimetres + 96 cubic centimetres + 96 cubic centimetres + 64 cubic centimetres = 416 cubic centimetres.

2) Volume of the given figure = Volume of vertical cuboid + Volume of vertical cuboid + Volume of cube, Length = 16 centimetre, Breadth = 4 centimetre, Height = 3 centimetre, Volume of vertical cuboid = length X breadth X height = 16 X 4 X 3 = 192 cubic centimetres, Length of side of cube = 4 centimetres, Volume of cube = side X side X side = 4 X 4 X 4 = 64 cubic centimetres, Volume of the given figure = 192 cubic centimetres + 192 cubic centimetres + 64 cubic centimetres = 448 cubic centimetres.

3) Volume of the given figure = Volume of vertical cuboid + Volume of horizontal cuboid, Length = 11 centimetre, Breadth = 4 centimetre, Height = 3 centimetre, Volume of vertical cuboid = length X breadth X height = 11 X 4 X 3 = 132 cubic centimetres, Length = 16 centimetre, Breadth = 4 centimetre, Height =3 centimetre, Volume of horizontal cuboid = length X breadth X height = 16 X 4 X 3 = 192 cubic centimetres, Volume of the given figure = 132 cubic centimetres + 192 cubic centimetres = 324 cubic centimetres.

Large measures Textbook Page No. 67

What is the volume of a cube of side 1 metre?
I metre means 100 centimetres?
So, we must calculate the volume of a cube of side 100 centimetres. How much is it?
We say that the volume of cube of 1 metre is 1 cubic metre.
So,
1 cubic metre = 1000000 cubic centimetre.
Volume of large objects are often said as cubic metres.

Question 1.
A truck is loaded with sand, 4 metre long, 2 metre wide and 1 metre high. The price of 1 cubic metre of sand is 1000
rupees. What is the price of this truck load?
Price of the truck loaded with sand is 8000 rupees,

Explanation:
Volume of the truck loaded with sand =length X breadth X height = 4 X 2 X 1 = 8 cubic metres.
Price of 1 cubic metre of sand = 1000 rupees, Price of the truck loaded with sand = 8 cubic metres X 1000 rupees = 8000 rupees, Therefore, price of the truck loaded with sand is 8000 rupees.

Question 2.
What is the volume in cubic metres of a platform 6 metre long, 1 metre wide and 50 centimetre high?
Volume of the rectangular block is 3 cubic metres,

Explanation:
Length = 6 metre, Breadth = 1 metre, Height = 50 cm = $$\frac{1}{2}$$ metre, Volume of the rectangular block = length X breadth X height = 6 X 1 X $$\frac{1}{2}$$ = 3 cubic metres.

Question 3.
What is the volume of a piece of wood which is 4 metres long, $$\frac{1}{2}$$ metre wide and 25 centimetre high? The price of 1 cubic metre of wood is 60000 rupees. What is the price of this piece of wood?
Price of the piece of wood is 30,000 rupees,

Explanation:
Length = 4 metre, Breadth = $$\frac{1}{2}$$ metre, Height = 25 cm = $$\frac{1}{4}$$ metre,
Volume of a piece of wood = length X breadth X height = 4 X $$\frac{1}{2}$$ X $$\frac{1}{4}$$ = $$\frac{1}{2}$$ cubic metre, Price of 1 cubic metre of wood = 60000 rupees, Price of the piece of wood = $$\frac{1}{2}$$ cubic metre X 60000 rupees = 30000 rupees, Therefore, price of the piece of wood is 30,000 rupees.

Capacity

Look at this hollow box:

It is made with thick wooden planks. Because of the thickness, its inner length, breadth and height are less than the outer measurements.

The inner length, breadth and height are 40 centimetres, 20 centimetres and 10 centimetres.
So, a rectangular block of these measurement can exactly fit into the space within this box.
The volume of this rectangular block is the volume whithin the box.
This volume is called the capacity of the box.
Thus the capacity of this box is;
40 X 20 X 10 = 8000 cc
So, what is the capacity of a box whose inner length, breadth and height are 50 centimetres, 25 centimetres and 20 centimetres?
Capacity of the box is 25,000 cubic centimetres,

Explanation:
Length = 50 centimetres, Breadth = 25 centimetres, Height = 20 centimetres, Thus the capacity of the box = 50 X 25 X 20 = 25,000 cubic centimetres.

Litre and cubic metre

1 litre is 1000 cubic centimetres and 1 cubic metres is 1000000 cubic centimetres. So, 1 cubic metre is 1000 litres.

Liquid measures

What is the capacity of a cubical vessel of inner side 10 centimetres?
10 × 10 × 10 = 1000 cubic centimetres
1 litre is the amount of water that fills this vessel.
That is
1 litre = 1000 cubic centimetres
We can look at this in another way. Ifa cube of side 10 centimetres in completely immersed in a vessel, filled with water then the amount of water that overflows would be 1 litre.

So, how many litres of water does if a vessel of length 2 centimetres, breadth 15 centimetres and height 10 centimetres contain?

Let’s look at another problem:

A rectangular tank of length 4 metres and height 2$$\frac{1}{2}$$ metres can contain 15000 litres of water. What is the breadth of the tank?

If we find the product of length, breadth and height in metres. we get the volume in cubic metres.
Here the volume is given to be 15000 litres.
That is, 15 cubic metres.

The product of length and height is
4 X 2$$\frac{1}{2}$$ = 10
So, breadth multiplied by 10 is 15.
From this, we can calculate the width as $$\frac{15}{10}$$ = 1$$\frac{1}{2}$$ metre.

Now suppose this tank contains 6000 litres of water. What is the height of the water?
The amount of water is 6 cubic metres. So, the product of the length and breadth of the tank and the height of the water, all in metres is 6.
Product of length and breadth is; 4 X 1$$\frac{1}{2}$$ = 6
So, height is 6 ÷ 6 = 1 metre.

In the water Textbook Page No. 69

A vessel is filled with water. If a cube of side 1 centimetre is immersed into it, how many cubic centimetre of water would overflow? What if 20 such cubes are immersed?

If 1 cube is immersed then 1 cubic centimetre of water would overflow, If 20 cubes are immersed then 20 cubic centimetre of water would overflow,

Explanation:
Volume of the object added in the vessel = Volume of water displaced or overflowed, Volume of 1 small cube = side X side X side = 1 X 1 X 1 = 1 cubic centimetre. Therefore, 1 cubic centimetre of water were overflowed. Volume of 20 small cube = 20 (Volume of 1 small cube ) = 20 (side X side X side) = 20 (1 X 1 X 1) = 20 cubic centimetres. Therefore, 20 cubic centimetres of water were overflowed.

Raising water

A swimming pool is 25 metres long, 10 metres wide and 2 metre deep. It is half filled. How many litres of water does it contain now?
25 × 10 × 1 = 250 cubic metres
= 250000 litre
Suppose the water level is increased by 1 centimetre. How many more litres of water does it contain now?
25,2500 litres of water will be increased in the swimming pool,

Water level is increased by 1 centimetre, Length = 25 metres, Breadth = 10 metres, Height = 2.01 metres, Volume of swimming pool = length X breadth X height = 25 X 10 X 2.01 = 502.5 cubic metres = 502500 litres, Number of more litres of water to be added = Volume of swimming pool after increasing 1 centimetres – Volume of swimming pool = 502500 litres – 250000 litres = 25,2500 litres.

Textbook Page No. 70

Question 1.
The inner sides of a cubical box are of length 4 centimetres. What is its capacity? How many cubes of side 2 centimetres can be stacked inside it?
8 cubes can be stacked inside the cubical box,

Explanation:
Length = 4 centimetres, Volume of inner sides of a cubical box = length X length X length = 4 X 4 X 4 = 64 cubic centimetres. If side = 2 centimetres, Volume of the cube = side X side X side = 2 X 2 X 2 = 8 cubic centimetres, Cubes stacked inside the cubical box = $$\frac{Volume of inner sides of a cubical box}{Volume of the cube}$$ = $$\frac{64}{8}$$ = 8 cubes.

Question 2.
The inner side of a rectangular tank are 70 centimetres, 80 centimetres, 90 centimetres. How many litres of
water can it contain?
504 litres of water are in the rectangular tank,

Explanation:
Length = 70 centimetres, Breadth = 80 centimetres, Height = 90 centimetres, Capacity of the rectangular tank = 70 X 80 X 90 = 504000 cubic centimetres = 0.504 cubic metres = 504 litres ( since 1 cubic metre = 1000 litres).

Question 3.
The length and breadth of a rectangular box are 90 centimetres and 40 centimetres. It contains 180 litres of water. How high is the water level?
Height of the rectangular box is 50 centimetres,

Explanation:
Length = 90 centimetres, Breadth = 40 centimetres, Capacity of the rectangular box = 180 litres = 180000 cubic centimetres ( since 1 cubic centimetre= 0.001 litre), Volume of the rectangular box = Capacity of rectangular box = length X breadth X height = 90 X 40 X height = 180000 cubic centimetres, Product of length and breadth = 90 X 40 = 3600 square centimetres, Height of the rectangular box = $$\frac{180000}{3600}$$ = 50 centimetres.

Question 4.
The inner length, breadth and height of a tank are 80 centimetres, 60 centimetres and 15 centimetres, and it contains water 15 centimetre high. How much more water is needed to fill it?
The tank is already filled with 72 litres because the tank is filled 15 centimetres high,

Explanation:
Length = 80 centimetres, Breadth = 60 centimetres, Height = 15 centimetres, Capacity of the rectangular tank = 80 X 60 X 15 = 72000 cubic centimetres = 0.072 cubic metres = 72 litres ( since 1 cubic metre = 1000 litres).

Question 5.
The panchayat decided to make a rectangular pond. The length, breadth and depth were decided to be 20 metres, 15 metres and 2 metres. The soil dug out was removed in a truck which can cariy a load of length 3 metres, breadth 2 metres and height 1 metre. How many truck loads of soil have to be moved?
100 loads of soil have to be moved in the truck to form a rectangular pond,

Explanation:
Length = 20 metres, Breadth = 15 metres, Height = 2 metres, Volume of the rectangular pond = length X breadth X height = 20 X 15 X 2 = 600 cubic metres. If dimensions of truck are : Length = 3 metres, Breadth = 2 metres, Height = 1 metres, Volume of sand loaded in truck = length X breadth X height = 3 X 2 X 1 = 6 cubic metres. Number of truck loads of soil = $$\frac{Volume of the rectangular pond }{Volume of sand loaded in truck}$$ = $$\frac{600}{6}$$ = 100 loads.

Question 6.
The inner length and breadth of an aquarium are 60 centimetres and 30 centimetres. It is half filled with water. When a stone is immersed in it, the water level rose by 10 centimetres. What is the volume of the stone?
Volume of the stone is 18,000 cubic centimetres,

Explanation:
Length = 60 centimetres, Breadth = 30 centimetres, Height = h centimetres, Volume of the aquarium = length X breadth X height = 60 X 30 X h = 1800 h cubic centimetres. After immersing a stone then Length = 60 centimetres, Breadth = 30 centimetres, Height = (h+10) centimetres, Volume of the aquarium after adding a stone = length X breadth X height = 60 X 30 X (h+10) = 1800 (h+10) cubic centimetres, Volume of the stone = Volume of the aquarium after adding a stone – Volume of the aquarium = 1800 (h+10) – 1800 = 1800 h + 18000 – 1800 h = 18000 cubic centimetres.

Question 7.
A rectangular iron block has height 20 centimetres, breadth 10 centimetres and height 5 centimetres. It is melt and recast into a cube. What is the length of a side of this cube?
Length of side of the cube is 10 centimetres,

Explanation:
Length = 20 centimetres, Breadth = 10 centimetres, Height = 5 centimetres,
Volume of the rectangular iron block = length X breadth X height = 20 X 10 X 5 = 1000 cubic centimetres, The rectangular iron box is recast into a cube. Volume of the cube = Volume of the rectangular box = 1000 cubic centimetres, side X side X side = 1000 cubic centimetres, side of cube = 10 centimetres.

Question 8.
A tank 2$$\frac{1}{2}$$ metre long and 1 metre wide is to contain 10000 litres. How high must be the tank?
Height of the tank is 4 metres,

Explanation:
Length = 2.5 metres, Breadth = 1 metre, Capacity of the tank = 10000 litres =10 cubic metres ( since 1 cubic metre= 1000 litre), Volume of the tank = Capacity of the tank = length X breadth X height = 2.5 X 1 X height = 10 cubic metres, Product of length and breadth = 2.5 X 1 = 2.5 square metres, Height of the tank = 10 / 2.5 = 4 metres.

Question 9.
From the four corners of a square piece of paper of side 12 centimetres, small squares of side 1 centimetre are cut
off. The edges of this are bent up and joined to form a container of height 1 centimetre. What is the capacity of the container? If squares of side 2 centimetres are cut off, what would be the capacity?
Capacity of the container is 100 cubic centimetres, The capacity of square piece after cut off of squares is 84 cubic centimetres,

Explanation:
Dimensions of container are: Length = 12 – 1 – 1 = 10 centimetres, Breadth = 12 – 1 – 1 = 10 centimetres, Height = 1 centimetre, Capacity of the container = length X breadth X height = 10 X 10 X 1 = 100 cubic centimetres, If side of square = 2 centimetres, Capacity of the square = length X breadth X height = 2 X 2 X 1 = 4 cubic centimetres. Capacity of the square piece after cut  off of squares =  Capacity of the container – 4 (Capacity of the square) = 100 – 4(4) = 84 cubic centimetres.

## Kerala State Syllabus 6th Standard Maths Solutions Chapter 5 Decimal Forms

### Decimal Forms Text Book Questions and Answers

Measuring length Textbook Page No. 73

What is length of this pencil?
6 centimetres and 7 millilitres.
How about putting it in millimetres only? 67 millimetres.
Can you say it in centimetres only?
One centimetre means 10 millimetres.
Putting it the other way round, one millimetre is a tenth of a centimetre.
That is, $$\frac{1}{10}$$ centimetre
1 millilitre = $$\frac{1}{10}$$ centimetre
So, 7 millimetres is $$\frac{7}{10}$$ centimetres.
Now can’t you say the length of the pencil in just centimetres?
6 centimetres, 7 millilitres = 6$$\frac{7}{10}$$ centimetres.

We also write this as 6.7 centimetre. To be read 6 point 7 centimetre. It is called the decimal form of 6$$\frac{7}{10}$$ centimetres.

Like this, 7 centimetre, 9 millimetre is $$\frac{9}{10}$$ centimetre. And we write it as 7.9 centimetre in decimal form.

Now measure the length of your pencil and write it in decimal form.

My pencil is exactly 8 centimetres. How do I write it in decimal form?

Just write 8.0

Since in 8 centimetres, there is no millimetre left over, we may write it as 8.0 centimetres also.
Lengths less than one centimetre is put as only millimetres. How do we write such lengths as centimetres?
8.0 centimetres,

Explanation:
We write 8 centimetres in decimal form as 8.0 centimetres.

For example, 6 millimetres means $$\frac{6}{10}$$ centimetres and so we write it as 0.6 centimetres (read 0 point 6 centimetres)
Like this, 4 millimetre = $$\frac{4}{10}$$ centimetre = 0.4 centimetre.

Different measures

Lengths greater than one centimetre are usually said in metres. How many centimetres make a metre?
In reverse, what fraction of a metre is a centimetre?
1 centimetre = $$\frac{1}{100}$$ metre.

Sajin measured the length of a table as 1 metre and 13 centimetres. How do we say it in metres only?

13 centimetres means $$\frac{13}{100}$$ of a metre.
That is, $$\frac{13}{100}$$ metre.
1 metre and 13 centimetre means 1$$\frac{13}{100}$$ metre. We can write this us 1.13 metres in decimal form.

Like this,
3 metres, 45 centimetres = 3$$\frac{45}{100}$$ metre = 3.45 metres.
Now how do we write 34 centimetres in terms ola metre?
34 centimetre = $$\frac{34}{100}$$ metre = 0.34 metre.

Vinu measured the length of a table as 1 metre, 12 centimetres, 4 millimetres.
How do we say it in terms of a metre?
12 centimetres means 120 millimetres.
With 4 millimetres more, it is 124 millimetres.
1 millimetre is $$\frac{1}{100}$$ of a metre.
So, 124 millimetres = $$\frac{124}{100}$$ metre.
1 metre and 124 millimetre together is 1$$\frac{124}{100}$$ metre.
Its decimal form is 1.124 metre.
Thus 5 metre, 32 centimetres, 4 millimetres in decimal form is,
5 metre, 324 millilitre = 5$$\frac{324}{1000}$$ = 5.324 metre.

Millimetre and metre
1 m = 100 cm
1 cm = 10 mm
1 m = 1000 mm
So,
1 cm = $$\frac{1}{100}$$ m
1 mm = $$\frac{1}{10}$$ cm
1 mm = $$\frac{1}{1000}$$ m

We can write other measurements also in the decimal form.
One gram is $$\frac{1}{1000}$$ of a kilogram.
So, 5 kilograms and 315 grams we can write as 5$$\frac{315}{1000}$$ kilograms.
Its, decimal form is 5.3 15 kilograms.

Like this,
4 grams 250 milligrams = 4$$\frac{250}{100}$$ gram = 4.250 grams.
A millilitre is $$\frac{1}{1000}$$ litre.
So,
725 millilitre = $$\frac{725}{1000}$$ litre = 0.725 litre.

Write the following measurements in fractional and in decimal form.

Explanation:
Wrote the given measurements in fractional and in decimal form above as 1. 4 cm 3 mm in fractional form is as 1 cm = 10 mm so 4 X 10 mm + 3 mm = 40 + 3 = 43 mm or $$\frac{43}{1}$$ mm and in decimal form is 4.3 cm, 2. 5 mm in fractional form is as 1 mm = $$\frac{1}{10}$$ cm so it is $$\frac{5}{10}$$ cm and in decimal form it is 0.5 cm, 3. 10 m 25 cm in fractional form is as 1 cm = $$\frac{1}{100}$$ m so it is $$\frac{1025}{100}$$ m and in decimal form it is 10.25 m, 4. 2 kg 125 g in fractional form as 1 g = $$\frac{1}{1000}$$ kg is $$\frac{2125}{1000}$$ kg and in decimal form it is 2.125 kg, 5. 16 l 275 ml in fractional form as 1 ml = $$\frac{1}{1000}$$ l so it is $$\frac{16275}{1000}$$ l in decimal form it is 16.275 l, 6. 13l 225 ml in fractional form as 1 ml = $$\frac{1}{1000}$$ l so it is $$\frac{13225}{1000}$$ l in decimal form it is 13.225 l 7. 325 ml in fractional form is as 1 ml = $$\frac{1}{1000}$$ l so it is $$\frac{325}{1000}$$ l in decimal form it is 0.325 l.

In reverse Textbook Page No. 77

1.45 metre as a fraction is 1$$\frac{45}{1000}$$ metre.
How much in metre and centimetre?
1 metre 45 centimetre.
That is 145 centimetres.
So, 1.45 metre means 145 centimetres.
Like this, how about writing 0.95 metre in centimetre?
How much centimetre is this?
145 centimetre shirt, 95 centimetre pants,

Explanation:
Given 1.45 metre for a shirt, 0.95 metre for pants, So in centimetres 1 m is equal to 100 centimetre so 1.45 metre is 100 + 45 = 145 centimetre for shirt. Now 0.95 metre = 0.95 X 100 centimetre = 95 centimetre pants.

Next try converting 0.425 kilograms into grams?

0.425 kilograms = $$\frac{425}{1000}$$ kilograms = 425 gram.

Fill up the table.

Explanation:
Filled the given table as shown above,1) 3.2 cm in expanded form is as 1 cm = 10 mm so 3 X 10 mm + 2 mm = 32 mm and 1 mm = $$\frac{1}{10}$$ cm, the fraction form is 3$$\frac{2}{10}$$ cm, 2) 1 mm = $$\frac{1}{10}$$ cm so 7 mm = $$\frac{7}{10}$$ cm and in decimal form it is 0.7 cm, 3) 3.41 m in expanded form is as 1 m = 100 cm so 3 X 100 cm + 41 cm = 341 cm and 1 cm = $$\frac{1}{100}$$ m, the fraction form is 3$$\frac{41}{100}$$ cm, 4) $$\frac{62}{10}$$ m = 6.2 m and 1m = 100 cm so 6.2 m = 6.2 X 100 = 620 cm, 5) 5.346 kg in expanded form is as 1 kg = 1000 g so 5 X 1000 g + 346 g = 5346 g and 1 kg =1000 g, the fraction form is $$\frac{5346}{1000}$$ kg, 6) 1 kg = 1000 g and 1 g = $$\frac{1}{1000}$$ kg so 425 g in decimal form is 0.425 kg and in fractional form is $$\frac{425}{1000}$$ kg, 7) 2.375 l in expanded form is as 1 l = 1000 ml so 2 x 1000 ml + 375 ml = 2375 ml and 1 ml = $$\frac{1}{1000}$$ l, the fraction form is $$\frac{2375}{1000}$$ l, 8) 1.350 l in expanded form is as 1 l = 1000 ml so 1 X 1000 ml + 350 ml = 1350 ml and 1 ml = $$\frac{1}{1000}$$ l, the fraction form is $$\frac{1350}{1000}$$ l, 9) 1 l = 1000 ml and 1 ml = $$\frac{1}{1000}$$ l then $$\frac{625}{1000}$$ l in decimal form is 0.625 l and in expanded form 0.625 l X 1000 ml = 625 ml.

One fractions, many from

The heights of the children in a class are recorded. Ravi is 1 metre, 34 centimetre tall. This was written 1.34 metres. Naufal is 1 metre, 30 centimetres tall and this was written 1.30 metres.
30 centimetres means $$\frac{30}{1000}$$ metre. This can be written $$\frac{3}{10}$$ metre.
So, why not write Ravi’s height as 1.3 metres?

“Both are right,” the teacher said.
Since, $$\frac{3}{10}$$ = $$\frac{30}{100}$$, we can write the decimal form of $$\frac{3}{10}$$ as 0.3 or 0.30.
Then Ravi had a doubt: Since $$\frac{3}{10}$$ = $$\frac{300}{1000}$$, we can write, 30 centimetres as 0.300 metres.
“It is also right,” the teacher continued. How we write decimals is a matter of convenience.

For example, look at some lengths measured in metre and centimetre.
1 metre 25 centimetres
1 metre 30 centimetres
1 metre 32 centimetres
It is convenient to write these like this:
1.25 metre
1.30 metre
1.32 metre

If we measure millimetres also like this:
1 metre 25 centimetres 4 millimetres
1 metre 30 centimetres
1 metre 32 centimetres
It is better to write them as:
1.254 metre
1.300 metre
1.320 metre
Like this how can we write the decimal form of 2 kilogram, 400 gram?
What about 3 litres. 500 millilitres?
2.400 kilograms, 3.500 litres,

Explanation:
1 kilogram = 1000 grams so 400 grams = $$\frac{400}{1000}$$ kilogram then 2 kilogram + 0.400 kilograms = 2.400 kilograms, 1 litre = 1000 millilitres so 500 millilitres = $$\frac{500}{1000}$$ litre then 2 litres + 0.400 litres = 2.400 litres and 3 litres.500 millilitres = 3.500 litres.

Place value Textbook Page No. 80

We have seen how we can write various measurements as fractions and in decimal forms.
If we look at just the numbers denoting these measurements, we see that they are fractions with 10, 100, 1000 so on as denominators.

For example, just as we wrote 2 centimetres, 3 millimetres as 2$$\frac{3}{10}$$ and then as 2.3, we can write 2$$\frac{3}{10}$$ as 2.3, whatever, be the measurement.
That is, 2.3 is the decimal form of 2$$\frac{3}{10}$$.
Similarly, 4.37 is the decimal from of 4$$\frac{37}{100}$$
We can write
2$$\frac{3}{10}$$ = 2.3
4$$\frac{37}{100}$$ = 4.37
and so on.

On the otherhand, numbers in decimal form can be written as fractions:
247.3 = 247$$\frac{3}{10}$$ = 247 + $$\frac{3}{10}$$
The number 247 in this can be split into hundreds, tens and ones:
247 = (2 × 100) + (4 × 10) + (7 × 1)
So, we can write 247.3 as
247.3 = (2 × 100) + (4 × 10) + (7 × 1) + (3 × $$\frac{1}{10}$$)

First we write
247.39 = 247$$\frac{39}{100}$$ = 247 + $$\frac{39}{100}$$
Then split $$\frac{39}{100}$$ like this:
$$\frac{39}{100}$$ = $$\frac{30+9}{100}$$ = $$\frac{30}{100}$$ + $$\frac{9}{100}$$ = $$\frac{3}{10}$$ + $$\frac{9}{100}$$ = (3 × $$\frac{1}{100}$$) + (9 × $$\frac{1}{100}$$)
So, we can write 247.39 like this
247.39 = (2 × 100) + (4 × 10) + (7 × 1) + (3 × $$\frac{1}{10}$$) + (9 × $$\frac{1}{100}$$)
In general, we can say this:

In a decimal form, we put the dot to separate the whole number part and the fraction part. Digits to the left of the dot denote multiples of one. ten, hunded and so on; digits on the right denote multiples of tenth, hundredth, thousandth and so on.

For example. 247.39 can be split like this:

Can you split the numbers below like this.
1.42 16.8 126.360 1.064 3.002 0.007
Yes we can split the numbers,

Explanation:

1.42 = (1 X 1) + (4 X $$\frac{1}{10}$$) + (2 X $$\frac{1}{100}$$),

16.8 = (1 X 10) + (6 X 1) + (8 X $$\frac{1}{10}$$),

126.360 = (1 X 100) + (2 X 10) + (6 X 1) + (3 X $$\frac{1}{10}$$) + (6 X $$\frac{1}{100}$$),
1.064 = (1 X 1) + (0 X $$\frac{1}{10}$$) + (6 X $$\frac{1}{100}$$) + (4 X  $$\frac{1}{1000}$$),

3.002 = (3 X 1) + (0 X $$\frac{1}{10}$$) + (0 X $$\frac{1}{100}$$) + (2 X $$\frac{1}{1000}$$),

0.007 = (0 X 1) + (0 X $$\frac{1}{10}$$) + (0 X $$\frac{1}{100}$$) + (7 X $$\frac{1}{1000}$$).

Fraction and decimal Textbook Page No. 81

$$\frac{1}{2}$$ centimetres means, 5 millimetres. Its decimal form is 0.5 centimetre. So the decimal form of the fmction $$\frac{1}{2}$$ is 0.5
$$\frac{1}{2}$$ = $$\frac{5}{10}$$ right?
Similarly, what is the decimal form of $$\frac{1}{5}$$?

Measurements again

Let’s look at the decimal form of some measurements again. For example, what is the decimal form of 23 metre, 40 centimetre. As seen earlier.
23 metre 40 centimetre = 23$$\frac{40}{100}$$ metre = 23.40 metre

Looking at just the numbers;
$$\frac{40}{100}$$ = $$\frac{4}{100}$$
23$$\frac{40}{100}$$ = 23$$\frac{4}{10}$$ = (2 X 10) + (3 X 1) + (4 X $$\frac{4}{10}$$) = 23.4
So, we can write 23 metre, 40 centimetre either as 23.40 metre or as 23.4 metre.
What about 23 metre, 4 centimetre?
23 metre 4 centimetres = 23$$\frac{4}{100}$$ metre
Writing just the numbers,
23$$\frac{4}{100}$$ = (2 X 10) + (3 X 1) + (4 X $$\frac{1}{100}$$)
= (2 X 10) + (3 X 1) + (0 X $$\frac{1}{10}$$) + (4 X $$\frac{1}{100}$$)
= 23.04
Here the 0 just after the dot shows that the fractional part of the number has no tenths (The 0 in 307 shows that, after 3 hundreds, this number has no tens, right?)

Thus we write 23 metres, 4 centimetres as 23.04 metres. How about 23 metres and 4 millimetres? 23 metres 4 millimetres
= 23$$\frac{4}{1000}$$ metres
Writing only the numbers,
23$$\frac{4}{1000}$$ = (2 X 10) + (3 X 1) + (4 X $$\frac{1}{1000}$$)
= (2 X 10) + (3 X 1) + (0 X $$\frac{1}{10}$$) + (0 X $$\frac{1}{100}$$) + (4 X $$\frac{1}{1000}$$)
= 23.004
Thus
23 metre 4 millimetres = 23.004 metre

Some other fractions

We cannot write $$\frac{1}{4}$$ as a fraction with denominator 10. But we have $$\frac{1}{4}$$ = $$\frac{25}{100}$$. So the decimal form of $$\frac{1}{4}$$ is 0.25. What is the decimal form of $$\frac{3}{4}$$? And $$\frac{3}{8}$$?
Decimal form of $$\frac{3}{4}$$ is 0.75 and decimal form of $$\frac{3}{8}$$ is 0.375,

Explanation:
Writing, $$\frac{3}{4}$$ = $$\frac{3}{4}$$ X $$\frac{25}{25}$$ = $$\frac{75}{100}$$ = 0.75 is the decimal form, $$\frac{3}{8}$$ = $$\frac{3}{8}$$ X $$\frac{125}{125}$$ = $$\frac{375}{1000}$$ = 0.375 is the decimal form.

Fill up this table.

Explanation:
1) 45 cm in fraction form is $$\frac{45}{100}$$ m because 1 m = 100 cm and in decimal form it is 0.45 m, 2) 315 g in fractional form is $$\frac{315}{1000}$$ kg because 1 kg = 1000 g and in decimal form it is 0.315 g, 3) 455 ml in fractional form is $$\frac{455}{1000}$$ ml because 1 l = 1000 ml and in decimal form it is 0.455 l, 4) $$\frac{5}{100}$$ m in decimal form is 0.05 m while in measurement it is 5 centimetres because 1 m = 100 cm, 5) $$\frac{42}{1000}$$ kg in decimal form is 0.042 kg and in measurement it is 42 grams because 1 kg = 1000 g, 6) 0.035 l in fractional form is $$\frac{35}{1000}$$ l and in measurement it is 35 ml because 1l = 1000 ml, 7) 3kg 5g in fractional form can be written as $$\frac{3005}{1000}$$ kg because 1kg = 1000 g so 3kg 5g = [(3 x 1000) + 5] g = 3005 g and in decimal form it is 3.005 g, 8) 2l 7ml in fractional form can be written as $$\frac{2007}{1000}$$ l because 1l = 1000 ml so 2l 7 ml = [(2 X 1000) + 7] ml = 2007 ml and in decimal form it is 2.007 ml, 9)3m 4cm in fractional form can be written as $$\frac{304}{100}$$ m because 1m = 100 cm so 3m 4cm = [(3 X 100) + 4] = 304 cm and in decimal form it is 3.04 m, 10) 3m 4mm in fractional form can be written as $$\frac{3004}{1000}$$ m because 1m = 100 cm and 1mm = 0.1 cm and in decimal form it is 3.004 m, 11) 4kg 50g in fractional form can be written as $$\frac{4050}{1000}$$ kg because 1 kg = 1000 g so 4kg 50g = [(4 x 1000) + 50] g = 4050g and in decimal form it is 4.050 g, 12) 4kg 5g in fractional form can be written as $$\frac{4005}{1000}$$ kg because 1 kg = 1000 g so 4kg 5g = [(4 X 1000) + 5] g = 4005g and in decimal form it is 4.005 g, 13) 4kg 5mg in fractional form can be written as $$\frac{4000005}{1000}$$ kg because 1 kg = 1000 g , 1 mg = 0.001 g so 4kg 5mg = [(4 X 1000) + 0.005] g = 4000.005 g and in decimal form it is 4.000005 g,14) 2ml in fractional form is $$\frac{2}{1000}$$ ml because 1 l = 1000 ml and in decimal form it is 0.002 l, 15) 0.02l in fractional form is $$\frac{20}{1000}$$ l because 1 l = 1000 ml and in measurement form it is 20 ml, 16) $$\frac{200}{1000}$$ l in decimal form is 0.2l and in measurement form it is 200 ml because 1l = 1000 ml.

More and less

Sneha’s height is 1.36 metre and Meena’s height is 1.42 metre. Who is taller?
In the sports meet, Vinu jumped 3.05 metres and Anu, 3.5 metres. Who won?
Vinu jumped 3 metres, 5 centimetres and Anu jumped 3 metres, 50 centimetres, right? So who won?
Meena is taller,
Anu Won,

Explanation:
Given Sneha’s height is 1.36 metre and Meena’s height is 1.42 metre. Now if we compare Sneha’s height and Meena’s height 1.42 metre is more or greater than 1.36 metre so Meena is taller,
Now given Vinu jumped 3 metres, 5 centimetres and Anu jumped 3 metres, 50 centimetres, Comparing Vinu and Anu jumped heights if we see 3 metres 50 centimetres is more or greater than 3 metres 5 centimetres so Anu jumped more height therefore Anu won.

Largest number
Which is the largest number among 4836, 568,97? What about these? 0.4836, 0.568, 0.97
We can also look at it like this. Both numbers have 3 in one’s place. The number 3.05 has zero in the tenth’s place while 3.50 has 5 in the tenth’s place. So 3.50 is the larger number.
Similarly which is the largest among 2.400 kilogram, 2.040 kilogram, 2.004 kilogram?
What about 0.750 litre and 0.075 litre.
2.400 kilogram is largest among the given numbers, 0.750 litre is largest among the given numbers,

Explanation:
The numbers have 4 in tenth’s place. The number 2.040 has zero in the tenth’s place while 2.004 has 4 in the thousandth place. So 2.400 is the larger number. The numbers have 7 in tenth’s place. The number 0.075 has zero in the tenth’s place . So 0.750 is the larger number.

Textbook Page No. 84

Question 1.
Find the larger in each of the pairs given below:
i) 1.7 centimetre, 0.8 centimetre
1.7 centimetre is larger,

Explanation:
In 1.7, 1 is in the one’s place while in 0.8, 0 is in the one’s place. So 1.7 is larger.

ii) 2.35 kilogram, 2.47 kilogram
2.47 kilogram is larger,

Explanation:
In 2.47, 4 is in the tenth’s place while in 2.35, 3 is the tenth’s place. So 2.47 is larger.

iii) 8.050 litre, 8.500 litre
8.500 litre is larger,

Explanation:
In 8.500 , 5 is in tenth’s place while in 8.050, 0 is in tenth’s place. So 8.500 is larger.

iv) 1.005 kilogram, 1.050 kilogram
1.050 kilogram is larger,

Explanation:
In 1.050, 5 is in the hundredth place while in 1.005, 0 is in the hundredth place. So 1.050 is larger.

v) 2.043 kilometre, 2.430 kilometre
2.430 kilometre is larger,

Explanation:
In 2.430, 4 is in tenth’s place while in 2.043, 0 is in tenth’s place. So 2.430 is larger.

vi) 1.40 metre, 1.04 metre
1.40 metre is larger,

Explanation:
In 1.40, 4 is in the tenth’s place while in 1.04, 0 is in the tenth’s place. So 1.40 is larger.

vii) 3.4 centimetre, 3.04 centimetre
3.4 centimetre is larger,

Explanation:
In 3.4, 4 is in the tenth’s place while in 3.04 , 0 is in the tenth’s place. So 3.4 is larger.

viii) 3.505 litre, 3.055 litre
3.505 litre is larger,

Explanation:
In 3.505, 5 is in the tenth’s place while in 3.055 , 0 is in the tenth’s place. So 3.505 is larger.

Question 2.
Arrange each set of numbers below from the smallest to the largest.
i) 11.4, 11.45, 11.04, 11.48, 11.048
11.04, 11.048, 11.4, 11.45, 11.48,

Explanation:
The number 11.4 has 4 in tenth’s place, the number 11.45 has 4 in tenth’s place, the number 11.04 has 0 in tenth’s place, the number 11.48 has 4 in tenth’s place, the number 11.048 has 0 in tenth’s place, comparing 11.04 and 11.048 we understand 11.04 has 0 in thousandth place while 11.048 has 8 so 11.04 is the smallest, among 11.4, 11.45, 11.48, 11.4 is least because 11.4 has 0 in it’s hundredth place while 11.45 and 11.48 have 5 and 8 respectively we know 8 is greater than 5 hence 11.48 is greater than 11.45. Therefore the order of numbers is 11.04, 11.048, 11.4, 11.45, 11.48.

ii) 20.675, 20.47, 20.743, 20.074, 20.74
20.074, 20.47, 20.675, 20.74, 20.743,

Explanation:
The number 20.074 has 0 in it’s tenth place while the other given numbers don’t have so it the smallest of all given numbers, 20.47 is the next smallest number because it has 4 in it’s tenth place while others have 6 and 7 in tenth place we know 7 is greater than 6 therefore 20.675 is the next least number, among both 20.74 and 20.743 we find 4 in hundredth place while 3 in thousandth place in one number and 0 in other so 20.743 is greater than 20.74. Therefore the order of numbers is 20.074, 20.47, 20.675, 20.74, 20.743.

iii) 0.0675, 0.064, 0.08, 0.09, 0.94
0.064, 0.0675, 0.08, 0.09, 0.94,

Explanation:
All the given numbers contain 0 in their tenth place so we need to check the hundredth place 0.064 and 0.0675 are the numbers with least hundredth place value when we check the thousandth place 7 is greater than 4 so 0.064 is the least and 0.0675 is the next least, among 0.08, 0.09, 0.094; 0.08 is least as 8 is less than 9 and when we compare 0.09 and 0.094 we find 0.09 has 0 in it’s thousandth place while 0.094 has 4 so 0.09 is least compared to 0.094. Therefore the order of numbers is 0.064, 0.0675, 0.08, 0.09, 0.94.

Which of 11.4, 11.47, 11.465 the largest?
We can write 11.4 as 11.400 and 11.47 as 11.470.
Now can’t we find the largest?
11.4, 11.465, 11.47,

Explanation:
All the given numbers have 4 in their tenth place so 11.4 is least because the hundredth place of the number is 0 when hundredth place is compared between  11.465 and 11.47 then 11.47 is greater than 11.465 as 7 is greater than 6. Therefore the order of numbers is 11.4, 11.465, 11.47.

A 4.3 centimetre long line is drawn and then extended by 2.5 centimetres.

What is the length of the line now?
We can put the length in millimetres and add
4.3 cm = 43 mm
2.5 cm = 25 mm

Total length 43 +25 = 68mm
Turning this back into centimetres, we get 6.8 centimetres.
We can do this directly, without changing to millimetres.

What if we want to add 4.3 centimetres and 2.8 centimetres?
If we change into millimetres and add, we get 71 millimetres.
And turing back into centiemtes, it becomes 7.1 centimetres.

Can we do this also directly, without changing to millimetres?
Let’s add in terms of place value.

The answer is 6 ones and 11 tenths; that is, 7 ones and
1 tenth. This we can write 7.1

How do we add 4.3 metres and 2.56 metres?
We can change both to centimetres and add

4.3 m = 430 cm
2.56 m = 256 cm
The length is 430 + 256 = 686 centimetres.
Changing back to metres, it is 6.86 metres.

We can add directly, without changing to centimetres 4.30
(when we do this, it is convenient to write 4.3 as 4.30)
What if we want to add 4.3 metres and 2.564 metre?
We can change both to millimetres and add
4300 mm + 2564 mm = 6864 mm
6864 mm = 6.864 mm = 6864

Generally speaking, to add measurements given in decimal form, it is better to make the number of digits in the decimal parts same; for this, we need only add as many zeros as needed.
Now look at this; if from a 12.4 centimetre long stick, a 3.2 centimetre piece is cut off, what is the length of the remaining part?

3 centimetres subtracted from 12 centimetres is 9 centimetres.
2 millimetres subtracted from 4 millimetres is 2 millimetres.
We can write it like this;

How do we subtract 3.9 centimetres from 15.6 centimetres?
We cannot subtract 9 millimetres from 6 millimetres. So we look at 15.6 centimetres as 14 centimetres and 16 millimetre. 9 millimetres subtracted from 16 millimetres gives 7 millimetres.
Let’s write according to place values and subtract.

Another example: A sack contains 16.8 kilograms sugar. From this, 3.750 kilogram is put in a bag. How much sugar remains in the sack? Write 16.8 kilogram as 16.8000 kilograms and try it.
Sugar remains in the sack is 13.050 kilograms,

Explanation:
Given a sack contains 16.8 kilograms sugar. From this, 3.750 kilogram is put in a bag. So sugar remains in the sack we wrote 16.8 kilogram as 16.8000 kilograms and subtracted 3.750 kilograms we get 13.050 kilograms.

Textbook Page No. 87

Question 1.
Sunitha and Suneera divided a ribbon between them. Sunitha got 4.85 metre and Suneera got 3.75 metre. What was the length of the original ribbon?
Total length of the ribbon is 8.60 metre,

Explanation:
Length of ribbon Sunitha has = 4.85 metre, Length of ribbon Suneera has = 3.75 metre, Total length of ribbon = 4.85 metre + 3.75 metre = 8.60 metre.

Question 2.
The sides of a triangle are of lengths 12.4 centimetre, 16.8 centimetre, 13.7 centimetre. What is the perimeter of the triangle?
Perimeter of the triangle is 42.9 centimetre,

Explanation:
Given sides of a triangle are 12.4 centimetre, 16.8 centimetre, 13.7 centimetre, Perimeter of the triangle = 12.4 centimetre + 16.8 centimetre + 13.7 centimetre = 42.9 centimetre.

Question 3.
A sack has 48.75 kilograms of rice in it. From this 16.5 kilograms was given to Venu and 12.48 kilograms to Thomas. How much rice is now in the sack’?
19.77 kilograms of rice is left in the sack,

Explanation:
Total quantity of rice the sack contain = 48.75 kilograms, Quantity of rice Venu was given = 16.5 kilograms, Quantity of rice Thomas was given = 12.48 kilograms, Quantity of rice left in the sack = Total quantity of rice the sack contain – ( Quantity of rice Venu was given + Quantity of rice Thomas was given) = 48.75 kilograms – (16.5 kilograms + 12.48 kilograms) = 19.77 kilograms.

Question 4.
Which number added to 16.254 gives 30?
13.746,

Explanation:
The number added to 16.254 to get 30 = 30 – 16.254 = 13.746.

Question 5.
Faisal travelled 3.75 kilometres on bicycle, 12.5 kilometres in a bus and the remaining distance on foot. He travelled 17 kilometres in all. What distance did he walk?
Distance walked by Faisal is 0.75 kilometres,

Explanation:
Total distance travelled by Faisal = 17 kilometres, Distance travelled by Faisal on bicycle = 3.75 kilometres, Distance travelled by Faisal on bus = 12.5 kilometres, Distance walked by Faisal = Total distance travelled by Faisal – (Distance travelled by Faisal on bicycle + Distance travelled by Faisal on bus) = 17 kilometres – (3.75 kilometres + 12.5 kilometres) = 0.75 kilometres.

Quantities of some items are written using fraction.
Onion 1$$\frac{2}{5}$$ kilogram
Tomato 1$$\frac{3}{4}$$ kilogram
Chilly $$\frac{1}{4}$$ kilogram
How much is the total weight? Do it by writing in decimal form which way is easier?
Total weight is 3.4 kilogram,

Explanation:
1kg = 1000 grams, $$\frac{2}{5}$$ X 1000 = 400 grams = 0.4 kilogram, $$\frac{3}{4}$$ X 1000 = 750 grams = 0.75 kilogram, $$\frac{1}{4}$$ = 250 grams = 0.25 kilogram, Weight of onions = 1$$\frac{2}{5}$$ kilogram = 1.4 kilogram, Weight of tomatoes = 1$$\frac{3}{4}$$ kilogram = 1.75 kilogram, Weight of chilly = $$\frac{1}{4}$$ kilogram = 0.25 kilogram,
Total weight of vegetables = 1.4 kilogram + 1.75 kilogram + 0.25 kilogram = 3.4 kilogram,

Question 6.
Mahadevan’s home is 4 kilometre from the school. He travels 2.75 kilometre of this distance in a bus and the remaining on foot. What distance does he walk?
Distance walked by Mahadevan is 1.25 kilometre,

Explanation:
Total distance from Mahadevan’s home to school = 4 kilometre, Distance travelled by bus = 2.75 kilometre, Distance walked = Total distance from Mahadevan’s home to school – Distance travelled by bus = 4 – 2.75 = 1.25 kilometre.

Question 7.
Susan bought a bangle weighing 7.4 grams and a necklace weighing 10.8 grams. She bought a ring also and the total weight of all three is 20 grams. What is the weight of the ring?
1.8 grams is the weight of the ring,

Explanation:
Total weight of three objects = 20 grams, Weight of bangle = 7.4 grams, Weight of necklace = 10.8 grams, Weight of ring = Total weight of three objects – (Weight of bangle + Weight of necklace) = 20 – (7.4 + 10.8) = 1.8 grams.

Question 8.
From a 10.5 metre rod, an 8.05 centimetre piece is cut off. What is the length of the remaining piece?
Length of the remaining piece is 10.4195 metre,

Explanation:
Total length of the rod = 10.5 metre, Length of piece cut off = 8.05 centimetre = 0.0805 metre ( since 1 metre = 100 centimetre), Length of remaining piece = Total length of the rod – Length of piece cut off = 10.5 – 0.0805 = 10.4195 metre.

Question 9.
We add 10.864 and the number got by interchanging the digits in its tenth’s and thousand’s this place. What do we get? What is the difference of these two numbers?
Sum of the given number with the interchanged number is 21.332, Difference between these two numbers is 0.396,

Explanation:
The given number = 10.864, The number obtained by interchanging the digits in its tenth’s and thousand’s place = 10.468, Sum of these numbers = 21.332,

Difference of these two numbers = 0.396,

Question 10.
When 12.45 is added to a number and then 8.75 subtracted, the result was 7.34. What is the original number?
The original number is 3.64,

Explanation:
Let the original number be x, when 12.45 is added to x and then subtracted by 8.75 we get 7.34, The original number ‘x’ is 12.45 + x – 8.75 = 7.34 then x = (7.34 + 8.75) – 12.45 = 16.09 – 12.45 = 3.64, Therefore the original number is 3.64.

## Kerala State Syllabus 6th Standard Maths Solutions Chapter 6 Numbers

### Numbers Text Book Questions and Answers

Let’s make a rectangle Textbook Page No. 95

A rectangle with 20 dots:

5 dots wide, 4 dots high.
Can we make other rectangles, rearranging the dots?

Also like this:

Any other?
What about the number of dots along the width and height?
Their product must be 20, right?
In what all ways can we write 20 as the product of two natural numbers?
4 and 5 are the two natural numbers.
Explanation:
4 x 5 = 20
5 x 4 = 20

Now make different rectangles with 24 dots. Also write down the number of dots along the width and height.

Explanation:
Total number of 24 dots,
2, 4, 6 and 12 are factors of 24,
and the product of 4 x 6 = 24 or 6 x 4 =24
or 2 x 12  or 12 x 2 = 24

Let’s think about it, without actually making rectangles. What are the possible number of dots along the width and height?
The product of numbers in every row of the table is 30.

Explanation:
5 x 6 = 30 or 6 x 5 = 30
5 dots horizontal by 6 dots vertical.
There’s another way of stating this; all these numbers are factors of 30.

Now can you write down the different rectangles with 40 dots?
And 60 dots?

Explanation:
9 x 5 = 45
5 x 9 = 45
10 x 6 = 60
6 x 10= 60
61 not possible

Factor pairs

What are the factors of 72?
Two quick ones are 1 and 72.
We can divide 72 by 2 without any remainder. That is 2 is also a factor of 72.And 72 divided by 2 gives 36.
72 = 2 × 36
So 36 is also a factor of 72.
Thus we can find factors in pairs.
Since
72 ÷ 3 = 24
We have
72 = 3 × 24
This gives 3 and 24 as another pair of factors.

Can’t we find other pairs like this?

(1, 72)

(2, 36)

(3, 24)

(4, 18)

(6, 12)

(8, 9)

Now try to find the factors of 90, 99, 120 as pairs.
Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Factors of 99 = 1, 3, 9, 11, 33, and 99.
Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.
Explanation:
A factor is a number that divides the given number without any remainder.
The factors of a number can either be positive or negative.
factor pairs of 90
(1, 90)
(2, 45)
(3, 30)
(5, 18)
(6, 15)
(9, 10)
factor pairs of 99
(1, 99)
(3, 33)
(9, 11)
factor pairs of 120
(1, 120)
(2, 60)
(3, 40)
(4, 30)
(5, 24)
(6, 20)
(8, 15)
(10, 12)

• If 2 and 3 are factors of a number, should 6 also be a factor of that number’?
yes,
Explanation:
Yes, 6 also be a factor of that number.
2 x 3 = 6
6 x 1 = 6
3 x 2 = 6

• If 3 and 5 are factors of a number, should 15 also be a factor of that number’?
yes,
Explanation:
5 x 3 = 15
15 x 1 =15
3 x 5 = 15
3, 5 are the factors of 15.

• If 4 and 6 are factors of a number, should 24 also be a factor of that number’?
Yes,
Explanation:
4 x 6 = 24
6 x 4 = 24
4 , 6 are the factors of 24

• If 4 and 6 are factors of a number, what is the largest number we can say for sure is a factor of that number’?
12,
Explanation:
12 is the largest number is a factor of 24.
The largest number is 12.
The numbers whose factors and 4 and 6 are common multiples of 4 and 6 .
These numbers are also multiples of common multiple of 12.

• Given Two factors of a number, under what conditions can we say for sure that the product of these factors is also a factor’?
2 and 12.
Explanation:
2 x 12 = 24
The numbers whose factors and 2 and 12 are common multiples of 2 and 12.
These numbers are also multiples of common multiple of 24.

Odd and even

We have found the factors of many numbers like 20, 24, 30, 40, 45, 60, 61, 72, 90, 99, 120.
See how many factors each has.
All of them have an even number of factors, right?
Why is this so?
Is it true for all numbers?
Write the factor pairs of 36.
(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
So what are the factors of 36?
1, 2, 3, 4, 6, 9, 12, 18, 36
9 factors in all.

Why is the number of factors odd in the case?
Can you find any other number with an odd number of factors?
Take 16, for example.
What is the specialty of numbers with an odd number of factors?
1, 5, 25
Explanation:
1 x 25 = 25
5 x 5 =25
1, 5, 25 are the factors of 25.

Can you find all numbers between 1 and 100; which have an odd number of factors?
Yes,
Explanation:
To find an odd factor, exclude the even prime factor 2.
The odd numbers from 1 to 100 are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99

Repeated multiplication Textbook Page No. 97

How many factors does 5 have?
5 and 17 are prime numbers, aren’t they?
And a prime has only two factors, right? 1 and the number itself.

All composite numbers have more than two factors.
For example, let’s have a look at 32.

32 = 2 × 2 × 2 × 2 × 2
Taking the first 2 alone and all the other 2’s together, we can write
32 = 2 × 16
How about taking the first two 2 s together and then other 2’s together?
32 = 4 × 8
Taking all the 2’s together can be written as
32 = 1 × 32
Thus, the factors of 32 are the 6 numbers
1, 2, 4, 8, 16, 32
Let’s look at the factors of 81 like this:
Writing 81 as a product of prime numbers, we get
81 = 3 × 3 × 3 × 3
So we can write 81 as
3 × 27
9 × 9
1 × 81
Thus we have five factors.
1, 3, 9, 27, 81
We can put this in a different way.
Taking 3’s in groups we get the factors.
3
3 × 3 = 9
3 × 3 × 3 = 27
3 × 3 × 3 × 3 = 81
and find the 5 factors of 81 as 1, 3, 9, 27 and 81.
In these examples, 32 is a product of 2’s; and 81 is a product of 3’s.

Like this, can’t we easily find the factors of a number, which can be factorized as repeated product of a single prime?
This is true as any composite number can be expressed in terms of prime factors.
Explanation:
The prime factor for number 2 is same for the numbers.
for example;
12 = 2 x 2 x 3
18 = 2 x 3 x 3
both have prime factors 2 and 3 but the combination 12 is different from that of 18.

We can split 216 as
216 = 6 × 6 × 6
Can we say that the only factors of 216 are the 4 numbers 1, 6, 36, 24. What are the other factors of 216?

Therefore, the factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
Explanation:
The factor of a number is a number that divides the given number without any remainder.
To find the factors of given number, divide the number with the least prime number, i.e. 2.
1 x 216 =216
2 x 108 = 216
3 x 72 = 216
4 x 54 = 216
6 x 36 = 216
8 x 27 = 216
9 x 24 = 216
12 x 18 = 216

Question 1.
Find all the factors of the numbers below:
(i) 256
The factors of 256 are 1, 2, 4, 8, 16, 32, 64, 128 and 256.
Explanation:
The factors of 256 are 1, 2, 4, 8, 16, 32, 64, 128 and 256.
1 x 256 =256
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 x 128 = 256
2 x 2 x 64 = 4 x 64 = 256
2 x 2 x 2 x 64 = 8 x 32 = 256
2 x 2 x 2 x 2 x 16 = 16 x 16 = 256

(ii) 625
The factors of 625 are 1, 5, 25, 125 and 625.
Explanation:
The factors of 625 are 1, 5, 25, 125 and 625.
1 x 625 = 625
5 x 125 = 625
25 x 25 = 625

(iii) 243
The factors of 243 are 1, 3, 9, 27, 81 and 243.
Explanation:
The factors of 243 are 1, 3, 9, 27, 81 and 243.
3 x 81 = 243
9 x 27 = 243

(iv) 343
The factors of 343 are 1, 7, 49 and 343.
Explanation:
The factors of 343 are 1, 7, 49 and 343.
1 x 343 = 343
7 x 49 = 343

(v) 121
The factors of 121 are 1, 11 and 121.
Explanation:
The factors of 121 are 1, 11 and 121.
1 x 121 = 121
11 x 11 = 121

Question 2.
Which are the numbers between 1 and 100 having exactly three factors?
4, 9 , 25 and 49.
Explanation:
1, 2 and 4 three factor for 4
1, 3 and 9 three factor for 9
1, 5 and 25 three factor for 25
1, 7 and 49 three factor for 49
We know that the numbers between 1 and 100,
which have exactly three factors are 4, 9, 25 and 49.

Prime factors Textbook Page No. 99

How do we find the factors of 16?
The only prime factor of 16 is 2. Writing
16 = 2 × 2 × 2 × 2
We see that the factors of 16, except 1, are products of 2’ s.
2
2 × 2 = 4
2 × 2 × 2 = 8
2 × 2 × 2 × 2 = 16
Taking 1 also, we get all the factors of 16 as 1,2, 4, 8, 16.
Now let’s try 16 × 3 = 48.
48 = (2 × 2 × 2 × 2) × 3
To get its factors, we can multiply some of the 2’s only; or some 2 ‘s and 3.
Taking only 2’s, what we get are the factors of 16.
2, 4, 8, 16
What if we take 2’s and 3?
(2 × 3) = 6
(2 × 2) × 3 = 4 × 3 = 12
(2 × 2 × 2) × 3 = 8 × 3 = 24
(2 × 2 × 2 × 2) × 3 = 48
Thus we get also the factors.
6, 12, 24, 48

3 alone is also a factor. Also 1,which is a factor of every number. We can separate these factors like this;

What is the relation between each number in the first row with the number below it.
Now let’s take 48 × 3 = 144
144 = (2 × 2 × 2 × 2) × (3 × 3)
The factors can be got by taking only some 2’ s, some 2’ s and one 3 or some 2’s and two 3’s.
Taking 3 ’s only we get 3 and 9.
And 1 also is a factor.

These can also be written in a table like this:
The numbers in the first row, multiplied by 3, give the numbers in the second row.
And numbers in the second row, multiplied by 3, give the numbers in the third row.

Let’s look at the table along the columns.
First column is 1, 3, 9; these numbers do not have 2 as a factor.
Second column is 2, 6, 18; these have a single 2 as a factor.
What about the third and fourth columns?

Thus the numbers in each column, multiplied by 2, give the numbers in the next column.
So, a factor of 144 can be found like this:

Multiply some 2’s and 3’s . The number of 2’s must be less than or equal to 4 (we can also choose to take no 2 at all). The number of 3’s must be less than or equal to 2 (or no 3 at all). Such factors, together with lgive all the factors.

For example, 24 is the product of three 2’s and one 3.
24 = 2 × 2 × 2 × 3
And 18 is the product of a single 2 and two 3’s.
Can you find the factors of 200 like this?
200 = 2 × 2 × 2 × 5 × 5
Make a table like this:

Explanation:
factors of 200 = 2 × 2 × 2 × 5 × 5
Thus the numbers in each column, multiplied by 5, to give the numbers in the next column.

Find all the factors of the numbers below:

(i) 242
1, 2 and 11 are the factors of 242
Explanation:
2 × 11 = 22
2 × (11 × 11) = 22 × 11 = 242

(ii) 225
1, 3 and 5 are the factors of 225
Explanation:
5 × 5 = 25
(5 x 5) × (3 × 3) = 25 × 9 = 225

(iii) 400
1, 2 and 5 are the factors of 400
Explanation:
5 × 5 = 25
2 x 2 x 2 x 2 = 16
(5 x 5) × (2 x 2 x 2 x 2) = 25 × 16 = 400

(iv) 1000
1, 2 and 5 are the factors of 400
Explanation:
5 × 5 x 5 = 125
(5 x 5 x 5) × (2 x 2 x 2) = 125 × 8 = 400

We have found the factors of 144.
Now let’s try 144 × 5 = 720
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
We can separate the factors as those without 5 and those with 5.
The factors without 5 are factors of 144.
And these can be found as before.

Multiplying all these by 5 gives the factors with 5.

Let’s write all these factors of 720 in a single table:

What about 144 × 25 = 3600?
We can expand the factor table of 720 like this:

Factorize each of the numbers below as the product of primes and write all factors in a table.
Write also the number of factors of each.

(i) 72
1, 2 and 3 are factors of 72

Explanation:
factorization of 72 is,
1 x 2 = 2
3 × 3 = 9
2 x 2 x 2  = 8
(3 x 3) × (2 x 2 x 2) = 9 × 8 = 72
So, 1, 2 and 3 are factors of 72.

(ii) 108

Explanation:
factorization of 108 is,
2 × 2 = 4
(3 x 3 x 3) × (2 x 2) = 27 × 4 = 108
So, 1, 2, 3 are factors of 108

(iii) 300

Explanation:
Factorization of 300 is,
3 x 2 x 2 x 5 x 5 x 1
= 3 x 4 x 25
= 300

(iv) 96
1, 2 and  3 are factors of 96

Explanation:
factorization of 96 is 3 x 2 x 2 x 2 x 2 x 2 x 1.
factors of 96 are 1, 2 and 3.

(v) 160

Explanation:
factorization of 160 is,
2 x 2 x 2 x 2 x 2 x 5 x 1 = 160

(vi) 486
2 x 3 x 3 x 3 x 3 x 3 = 486
Explanation:
factorization of 486 is,
2 x 3 x 3 x 3 x 3 x 3 = 486
LCM of 486 is as follows,

(vii) 60
1, 2 ,3 and 5 are the factors of 60

Explanation:
factorization of 60 is,
2 x 2 x 3 x 5 = 160
LCM of 60 is shown below,

(viii) 90

Explanation:
factorization of 90 is,
2 x 3 x 3 x 5 = 90
factor of 90 are 1, 2, 3 and 5
LCM of 90 is shown below,

(ix) 150

Explanation:
factorization of 150 is,
2 x 3 x 5 x 5 = 150
LCM of 150 is shown below,

(i) Find the number of factors of 6, 10, 15, 14, 21. Find some other numbers with exactly four factors.
Factors of 6 are 1, 2, 3 and 6.
Factors of 10 are 1, 2, 5 and 10.
Factors of 15 are 1, 3, 5 and 15.
Factors of 14 are 1, 2, 7 and 14.
Factors of 21 are 1, 3, 7 and 21.
Explanation:
A factor is a number which divides the number without remainder.
Factors of 6 are 1, 2, 3 and 6.
1 x 6 = 6
2 x 3 = 6
3 x 2 = 6
6 x 1 = 6
Factors of 10 are 1, 2, 5 and 10.
1 x 10 = 10
2 x 5 = 10
5 x 2 = 10
10 x 1 = 10
Factors of 15 are 1, 3, 5 and 15.
1 x 15 = 15
3 x 5 = 15
5 x 3 = 15
15 x 1 = 15
Factors of 14 are 1, 2, 7 and 14.
1 x 14 = 14
2 x 7 = 14
7 x 2 = 14
14 x 1 = 14
Factors of 21 are 1, 3, 7 and 21.
1 x 21 = 21
3 x 7 = 21
7 x 3 = 21
21 x 1 = 21

(ii) Is it correct to say that any number with exactly four factors is a product of two distinct primes?
No, this is not correct.
Explanation:
If any prime number then its 3rd power has exactly 4 divisors,
and is obviously not the product of two distinct primes.
Take 36, it has two prime factors, 2 and 3.

Number of factors Textbook Page No. 104

We know how to find all the factors of 64.
Without writing down all the factors, can we just find the number of factors?
64 = 2 × 2 × 2 × 2 × 2 × 2
We can take one 2, two 2’s, three 2’s and so on to get factors. How many such factors are there?
Here there are six 2’s. So we can take one to six 2’s, and 1 is also a factor.
6 + 1 = 7 factors in all.
Can we find the number of factors of243 like this?
243 = 3 × 3 × 3 × 3 × 3
How many 3’s?
Taking one 3, two 3’s, three 3’s and so on, how many factors do we get?
Together with 1?
5 + 1 = 6 factors in all.
If a number can be split as the repeated product of a single prime, how do we find the number of factors of that number quickly?
What if we have two primes?
For example, let’s take 64 × 3 = 192
192 = (2 × 2 × 2 × 2 × 2 × 2) × 3
1 and products of 2’s give 7 factors as above; these factors multiplied by 3 give another 7 factors. Altogether 14 factors.
So how many factors does 192 × 3 = 576 have?
576 = (2 × 2 × 2 × 2 × 2 × 2) × (3 × 3)

We can separate the factors of 576 like this.

(i) Factors without 3
1 2 4 8 16 32 64

(ii) Product of these by 3
3 6 12 24 48 96 192

(iii) Product of the first factors by two 3’s
9 18 36 72 144 288 576

7 of each type. 7 × 3 = 21 in all.
We can put this in a different way. Take the products of 2’s and 3’s separately.
576 = 64 × 9

Look at the three types of factors of 576 again

(i) 1, 2, 4, 8, 16, 32, 64 – Factors of 64
(ii) 3, 6, 12, 24, 48, 96, 192 – Products of the factors of 64 by the factor 3 of 9
(iii) 9, 18, 36, 72, 144, 288, 576 – Products of the factors of 64 by the factor 9 of 9

We can also say that the factors we write first are the product of the factors of 64 by the factor 1 of 9.

Thus the factors of 576 are the product of each factor of 64 by each factor of 9.
64 has 7 factors and 9 has 3 factors. So 64 × 9 = 576 has 3 groups of 7 factors.
That is 7 × 3 = 21 factors.
Like this, can we find how many factors 1000 has?
1000 = (2 × 2 × 2) × (5 × 5 × 5)

In this, 2 × 2 × 2 = 8 has 4 factors; and 5 × 5 × 5 = 125 also has 4 factors.
We can multiply each of the 4 factors of 8 by each of the 4 factors of 125 to get all factors of 1000. That is 4 groups of 4 factors, making 4 × 4 = 16 in all.
Now let’s see how many factors 3600 has.
3600 = (2 × 2 × 2 × 2) × (3 × 3) × (5 × 5)
2 × 2 × 2 × 2 = 16 has 5 factors, 3 × 3 = 9 and 5 × 5 = 25 have 3 factors each.

Multiplying each factor of 16 by each factor of 9 gives 5 × 3 = 15 factors of 16 × 9.
Multiplying each of these by factors of 25 give all factors of 16 × 9 × 25 = 3600.
That means 15 × 3 = 45 factors
(Look once more the factor table of 3600 done earlier).

The number 4 has 3 factors and number 6 has 4 factors. Can we say that 4 × 6 = 24 has 3 × 4 = 12 factors’? Multiply each factor of 4 by each factor of 6. Why did we get the number of factors wrong?
Yes, 4 × 6 = 24 has 3 × 4 = 12 factors’.
Explanation:
factors of 4 = 1, 2, 4
factors of 6 = 1, 2, 3, 6
Let 48 be the number,
factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
4 is one of the factor of 48,
6 is one of the factor of 48.
4 x 6 = 24 is also a factor of 48.
let x be the number,
if 4 and 6 are 2 of the factors of x,
then 4 x 6 = 24 is also one of the factor of x.

Textbook Page No. 107

Question 1.
The factor table of a number is given below. Some of the factors are written.

(i) What is the number with this factor table?
1, 2 , 14, 100, 245, 490, 4900
Explanation:
1 x 1 = 1
1 x 2 = 2
7 x 2 = 14
7 x 7 x 5 = 245
2 x 2 x 5 x 5 x 7 x 7 = 4900

(ii) Fill in the numbers in the circles
1, 14, 245, 4900
Explanation:
1 x 1 = 1
7 x 2 = 14
7 x 7 x 5 = 245
2 x 2 x 5 x 5 x 7 x 7 = 4900

(iii) Write the numbers below in the correct cells
4, 25, 140, 200

Explanation:
Take the LCM of given numbers 4, 25, 140 and 200
or find the factors of all the numbers above to fit in the cells as shown below.

(iv) Which of the numbers below cannot be in the table?
32, 40, 50, 200, 300, 350
32 and 300
can not be in the table
Explanation:
Due to non availability of factor Five 2’s and 3 in the table 32 and 300 can not be in the table

Question 2.
Find the number of factors of each of these numbers.
(i) 500
12
Explanation:
find the LCM of 500
500 = 2x 53.
= (2 + 1) x (3 + 1)
= 3 x 4
= 12

(ii) 600
24
Explanation:
find the LCM of 600
600 = 23 x 31 x 52.
= (3 + 1) x (1 + 1) x (1 + 2)
= 4 x 2 x 3
= 24

(iii) 700
12
Explanation:
first take the LCM of 700
700 = 22 x 52 x 71.
= (2 + 1) x (1 + 2) x (1 + 1)
= 3 x 3 x 2
= 12

(iv) 800
18
Explanation:
find the LCM of 800
800 = 25 x 52.
= (5 + 1) x (1 + 2)
= 6 x 3
= 18

(v) 900
27
Explanation:
Find the LCM of 900
900 = 22 x 32 x 52.
= (2 + 1) x (2 + 1) x (2 + 1)
= 3 x 3 x 3
= 27

Question 3.
How many factors does a product of three distinct primes have? What about a product of 4 distinct primes?
Product of 3 distinct primes have 8 factors.
Product of 4 distinct primes have 12 factors.
Explanation:
To find the factors of product of three distinct primes have and four distinct primes have.
The factors of product of three distinct primes have four factors that include 1.
Let us consider a product of the primes 2, 3 and 5.
2 × 3 × 5 = 30
Factors of 30 are 1 , 2 , 3, 5 , 6 , 10 , 30 where the prime factors are 2, 3 and 5.
Hence there are 8 factors in total for a number that is the product of three distinct primes.
Let us consider a product of the primes 2, 3, 5 and 7.
2 × 3 × 5 x 7 = 210
The factors of product of four distinct primes have five factors that include 1.
The factors of 210 are 1, 2, 3, 5, 6, 7, 15, 30, 42, 70, 105 and 210.
So, it has twelve factors.
Hence there are 8 factors in total for a number that is the product of three distinct primes.

Question 4.
i) Find two numbers with exactly five factors.
16 and 81
Explanation:
A factor is a number which divides the number without remainder.
factors of 16 = 1, 2, 4, 8 and 16.
factors of 81 = 1, 3, 9, 27, 81.

ii) What is the smallest number with exactly five factors?
16
Explanation:
A factor is a number which divides the number without remainder.
factors of 16 = 1, 2, 4, 8 and 16.

Question 5.
How many even factors does 3600 have?
36 factors.
Explanation:
Number of even factors = no. of total factors – no. of odd factors.
Prime Factorization of 3600 =  24 x 32 x 52
Total factors of 3600 = 45;
Total number of odd factors of 3600 is (2 + 1)(2 + 1) = 3 × 3 = 9
Even factors = 45 – 9 = 36

## Kerala State Syllabus 6th Standard Maths Solutions Chapter 7 Decimal Operations

### Decimal Operations Text Book Questions and Answers

Triangle Problem Textbook Page No. 109

Anup made a triangle with three sticks of length 4 centimetres each. What is the perimeter of this triangle?
How did you do it?
What is the perimeter?
4.3 + 4.3 + 4.3 = 12.9 cm.

Instead of adding again and again, we only compute 3 times 4.3
How do we find it?
4.3 centimetres mean 43 millimetres. And 3 times 43 millimetres is 43 × 3 = 129 millimetres.
This is 12.9 millimeters.
There’s another way of doing this:
4.3 = 4$$\frac{3}{10}$$ = $$\frac{43}{10}$$
So, 3 times $$\frac{43}{10}$$ is
$$\frac{43}{10}$$ × 3 = $$\frac{129}{10}$$ = 12.9 cm.
That is, 4.3 × 3 = 12.9

Cloth problem

To make a shirt for a boy in the class, 1.45 metres of cloth is needed, on average.
How much cloth is needed to make shirts for the 34 boys in the class?
We must calculate 34 times 1.45.
1.45 metres mean 145 centimetres; And 34 times 145 is
145 × 34 = 4930
How much metres is 4930 centimetres?
$$\frac{4930}{100}$$ metre = 49.30 metres

How about writing all measurements as tractions?
1.45 = 1$$\frac{45}{100}$$ = $$\frac{145}{100}$$
1.45 × 34 = 1$$\frac{45}{100}$$ × 34 = $$\frac{145}{100}$$ × 34 = $$\frac{4930}{100}$$
We can write it as a decimal.
$$\frac{4930}{100}$$ = 49.30 = 49.3
Thus 1.45 × 34 = 49.3

The area of a square of side 1 centimetre 1 square centimetre and the area of a square of side 1 millimetre is 1 square millimetre. 1 centimetre is 10 millimetres. So in the bigger square we can stack 10 smaller squares each along the length and breadth 10 × 10 = 100 small squares in all. So the smaller square is $$\frac{1}{100}$$ of the bigger square. That means 1 sq.mm. = $$\frac{1}{100}$$ sq.cm

Area

We know how to calculate the area of a rectangle of length 8 centimetres and height 6 centimetres. What about a rectangle of length 8.5 centimetres and breadth 6.5 centimetres? The lengths in millimetres are 85 and 65. So area is 85 × 65 = 5525 square millimetres. How do we change it into square centimetres?
1 square millimetre = $$\frac{1}{100}$$ square centimetre.
5525 square millimetres = $$\frac{5525}{100}$$ = 55.25 square centimetres.

How about writing all measurements as fractions?
8.5 centimetres = 8$$\frac{5}{10}$$ centimetres = $$\frac{85}{10}$$ centimetres
6.5 centimetres = 6$$\frac{5}{10}$$ centimetres = $$\frac{65}{10}$$ centimetres
Area is $$\frac{85}{10}$$ × $$\frac{65}{10}$$ square centimetres.
$$\frac{85}{10}$$ × $$\frac{65}{10}$$ = $$\frac{5525}{100}$$ = 55.25
Thus area is 55.25 square centimetres.
Let’s write the computation using numbers only.
8.5 × 6.5 = 55.25

Textbook Page No. 111

Question 1.
The sides of a square are of length 6.4 centimeters. What is its perimeter?
25.6 centimeters.
Explanation:
6.4 centimeter is length of one side,
The perimeter of a square is P = 6.4 + 6.4 + 6.4 + 6.4 = 25.6 centimeters.
There’s another way of doing this is,
Perimeter of a square is P = 4 x side
P = 4 x 6.4
P = 25.6 centimeters.

Question 2.
3 rods of length 6.45 meters each are laid end to end. What is the total length?
19.62 meters.
Explanation:
6.45 centimeter is length of one side.
The perimeter of a triangle is P = 6.45 + 6.45 + 6.45 = 19.62 meters.
There’s another way of doing this is,
Perimeter of a square is P = 3 x side
P = 3 x 6.45
P = 19.62 centimeters

Question 3.
A bag can be filled with 4.575 kilograms of sugar. How much sugar can be filled in 8 such bags?
36.6 kg
36.6 kilograms
Explanation:
A bag can be filled with 4.575 kilograms of sugar.
8 bags of sugar = 8 x 4.575 = 36.6 kg.

Question 4.
The price of one kilogram of rice is 34.50 rupees. How much money do we need to buy 16 kilograms?
552 rupees.
Explanation:
The price of one kilogram of rice is 34.50 rupees.
Cost of 16 kilograms = 16 x 34.50 = 552 rupees.

Question 5.
6 bottles are filled with the coconut oil in a can. Each bottle contains 0.478 liters. How much oil was in the can, in liters?
2.868 liters.
Explanation:
6 bottles are filled with the coconut oil in a can.
Each bottle contains 0.478 liters.
Total oil in the can, in liters was = 6 x 0.478 = 2.868 liters

Question 6.
The length and breadth of a rectangular room are 8.35 meters and 3.2 meters. What is the area of that room?
26.72 meter square.
Explanation:
Area of rectangle = length x breadth
length of a rectangle room = 8.35 meters
breadth of a rectangle room = 3.2 meters
Area = 8.35 x 3.2 = 26.72 meter square.

Multiplication

What is the meaning of 4.23 × 2.4?
4.23 × 2.4 = $$\frac{423}{100}$$ × $$\frac{24}{10}$$ = $$\frac{423 \times 24}{1000}$$
To compute this, we have to multiply 423 by 24 and then divide by 1000.
423 × 24 = 10152
$$\frac{423 \times 24}{1000}$$ × $$\frac{10152}{1000}$$ = 10.152

In the answer, how many digits are there after the decimal point? Why three?
Look at the fraction form of the answer. The denominator is 1000, right?
How did we get this 1000?

Look at the denominator of the fractions we multiplied.
So how do we complete 4.23 × 0.24?
First find 423 × 24 = 10152.

Now how many digits are there after the decimal point in the product?
If we write 4.23 × 0.24 as a fraction, what would be the denominator of the product?
4.23 as a fraction has denominator 100.
0.24 as a fraction has denominator 100.What about the denominator of the product?
So, 4.23 × 0.24 = $$\frac{10152}{10000}$$ = 1.0152

Like this, how do we do 2.45 × 3.72?
First calculate 245 × 372.
Now we must find out the number of digits after the decimal point.
What is the denominator of 2.45 as a fraction.
And of 3.72?
What is the denominator of the product?
So,
2.45 × 3.72 = 9.1140 = 9.114

0.1 × 0.1 = 0.01
0.01 × 0.01 = 0.0001
0.001 × 0.001 = 0.000001
0.0001 × 0.0001 = 0.00000001
Explanation:
To multiply a decimal number by a decimal number,
we first multiply the two numbers ignoring the decimal points.
Then place the decimal point in the product, in such a way that decimal places in the product is equal to the sum of the decimal places in the given numbers as shown above.

Textbook Page No. 113

Question 1.
Calculate the products below:

i) 46.2 × 0.23
10.626
Explanation:
If we write 46.2 × 0.23 as a fraction,
46.2 as a fraction has denominator 10.
0.23 as a fraction has denominator 100.
= $$\frac{462}{10}$$ × $$\frac{23}{100}$$
= $$\frac{462 \times 23}{1000}$$
To compute this, we have to multiply 462 by 23 and then divide by 1000.
462 × 23 = 10,626
= $$\frac{10626}{1000}$$ = 10.626
So, 46.2 × 0.23 = 10.626

ii) 57.52 × 31.2
1794.624
Explanation:
If we write 57.52 × 31.2 as a fraction,
57.52 as a fraction has denominator 100.
31.2 as a fraction has denominator 10.
= $$\frac{5752}{100}$$ × $$\frac{312}{10}$$
= $$\frac{5752 \times 312}{1000}$$
To compute this, we have to multiply 5752 by 312 and then divide by 1000.
5752 × 312 = 17,694,624
= $$\frac{17,94,624}{1000}$$ = 1794.624
So, 57.52 × 31.2 = 1794.624

iii) 0.01 × 0.01
Explanation:
If we write 0.01 × 0.01 as a fraction,
0.01 as a fraction has denominator 100.
0.01 as a fraction has denominator 100.
= $$\frac{1}{100}$$ × $$\frac{1}{100}$$
= $$\frac{1 \times 1}{10000}$$
To compute this, we have to multiply 1 by 1 and then divide by 10000.
1 × 1 = 1
= $$\frac{1}{10000}$$ = 0.0001
So, 0.01× 0.01 = 0.0001

iv) 2.04 × 2.4
4.896
Explanation:
If we write 2.04 × 2.4 as a fraction,
2.04 as a fraction has denominator 100.
2.4 as a fraction has denominator 10.
= $$\frac{204}{100}$$ × $$\frac{24}{10}$$
= $$\frac{204 \times 24}{1000}$$
To compute this, we have to multiply 204 by 24 and then divide by 1000.
204 × 24 = 4896
= $$\frac{4896}{1000}$$ = 4.896
So, 2.04 x 2.4 = 4.896

v) 2.5 × 3.72
9.3
Explanation:
If we write 2.5 × 3.72 as a fraction,
2.5 as a fraction has denominator 10.
3.72 as a fraction has denominator 100.
= $$\frac{25}{10}$$ × $$\frac{372}{100}$$
= $$\frac{25 \times 372}{1000}$$
To compute this, we have to multiply 25 by 372 and then divide by 1000.
25 × 372 = 9300
= $$\frac{9300}{1000}$$ = 9.3
So, 2.5× 3.72 = 9.3

vi) 0.2 × 0.002
0.0004
Explanation:
If we write 0.2 × 0.002 as a fraction,
0.2 as a fraction has denominator 10.
0.002 as a fraction has denominator 1000.
= $$\frac{2}{10}$$ × $$\frac{2}{1000}$$
= $$\frac{2 \times 2}{10000}$$
To compute this, we have to multiply 2 by 2 and then divide by 10000.
2 × 2 = 4
= $$\frac{4}{10000}$$ = 0.0004
So, 0.2 × 0.002 = 0.0004

Question 2.
Given that 3212 × 23 = 73876, find the products below, without actually multiplying?

i) 321.2 × 23 = _____
7387.6,
Explanation:
Given 3212 × 23 = 73876,
In 321.2 × 23 first find out the number of digits after the decimal point.
(1 + 0) = 1
So, 321.2 × 23 = 7387.6

ii) 0.3212 × 23 = _____
7.3876
Explanation:
Given 3212 × 23 = 73876,
In 0.3212 × 23 first find out the number of digits after the decimal point.
(4 + 0) = 4
So, 0.3212 × 23 = 7.3876

iii) 32.12 × 23 = ____
738.76
Explanation:
Given 3212 × 23 = 73876,
In 32.12 × 23 first find out the number of digits after the decimal point.
(2 + 0) = 2
So, 32.12 × 23 = 738.76

iv) 32.12 × 0.23 = ____
7.3867
Explanation:
Given 3212 × 23 = 73876,
In 32.12 × 0.23 first find out the number of digits after the decimal point.
(2 + 2) = 4
So, 32.12 × 0.23 = 7.3876

v) 3.212 × 23 = ____
73.876
Explanation:
Given 3212 × 23 = 73876,
In 3.212 × 23 first find out the number of digits after the decimal point.
(3 + 0) = 3
So, 3.212 × 23 = 73.876

vi) 321.2 × 0.23 = _____
73.876
Explanation:
Given 3212 × 23 = 73876,
In 321.2 × 0.23 first find out the number of digits after the decimal point.
(1 + 2) = 3
So, 321.2 × 0.23 = 73.876

Question 3.
Which of the products below is equal to 1.47 × 3.7?
i) 14.7 × 3.7
ii) 147 × 0.37
iii) 1.47 × 0.37
iv) 0.147 × 37
v) 14.7 × 0.37
vi) 0.0147 × 370
vii) 1.47 × 3.70
Option iv, v and vii has the equal products.
Explanation:
first find out the number of digits after the decimal point, then add to the product.
i) 14.7 × 3.7 = 54.39 (1 + 1 = 2)
ii) 147 × 0.37 = 54.39 (0 + 2 = 2)
iii) 1.47 × 0.37 = 0.5439 (2 + 2 = 4)
iv) 0.147 × 37 = 5.439 (3 + 0 = 3)
v) 14.7 × 0.37 = 5.439 (1 + 2 = 3)
vi) 0.0147 × 370 = 0.5439 (4 + 0 = 4)
vii) 1.47 × 3.70 = 0.5439 (2 + 2 = 4)
So, iv, v and vii has the equal products.

Question 4.
A rectangular plot is of length 45.8 meters and breadth 39.5 meters .What is its area?
1809.10 meter square.
Explanation:
Area of rectangle = length x breadth
length of a rectangle plot= 45.8 meters
breadth of a rectangle plot = 39.5 meters
Area = 45.8 x 39.5 = 1809.10 meter square.

Question 5.
The price of petrol is 68.50 rupees per liter. What is the price of 8.5 liters?
582.25 liters.
Explanation:
The price of petrol is 68.50 rupees per liter.
The price of 8.5 liters = 68.50 x 8.5 = 582.25 L

Question 6.
Which is the largest product among those below.
i) 0.01 × .001
ii) 0.101 × 0.01
iii) 0.101 × 0.001
iv) 0.10 × 0.001
Option (ii)
Explanation:
The largest product among those below are,
i) 0.01 × .001 = 0.00001 (2 + 3 = 5)
ii) 0.101 × 0.01 = 0.00101 (3 + 2 = 5)
iii) 0.101 × 0.001 = 0.000101 (3 + 3 = 6)
iv) 0.10 × 0.001 = 0.00010 (2 + 3 = 5)
So, the largest product is 0.101 x 0.01 = 0.00101

It is easy to calculate these products;
384 × 10
230 × 100

Now calculate these products:

• 125 × 10
1250
Explanation:
First multiply the number by ignoring zeros.
125 x 1 = 125
Then add zero to the product.
125 + 0 = 1250
So, 125 x 10 = 1250

• 4.2 × 10
42
Explanation:
To multiply a decimal by 10,
move the decimal point in the multiplication by one place to the right.
4.2 x 10 = 42

• 13.752 × 10
137.52
Explanation:
To multiply a decimal by 10,
move the decimal point in the multiplication by one place to the right.
13.752 x 10 = 137.52

• 4.765 × 100
476.5
Explanation:
To multiply a decimal by 100,
move the decimal point in the multiplication by two places to the right.
So, 4.765 x 100 = 476.5

• 3.45 × 100
345
Explanation:
To multiply a decimal by 100,
move the decimal point in the multiplication by two places to the right.
3.45 x 100 = 345

• 14.572 × 100
1457.2
Explanation:
To multiply a decimal by 100,
move the decimal point in the multiplication by two places to the right.
14.572 x 100 = 1457.2

• 1.345 × 1000
1345
Explanation:
To multiply a decimal by 1000,
move the decimal point in the multiplication by three places to the right.
1345 x 1000 = 1345

• 2.36 × 1000
0.236
Explanation:
To multiply a decimal by 1000,
move the decimal point in the multiplication by three places to the right.
2.36 x 1000 = 0.236

• 1.523 × 1000
1523
Explanation:
To multiply a decimal by 1000,
move the decimal point in the multiplication by three places to the right.
1.523 x 1000 = 1523

Have you found out an easy way to multiply decimals by numbers 10,100,1000 and so on?
Yes, just by moving the places towards the right.
Explanation:
When a decimal number is multiplied by 10, 100 or 1000,
the digits in the product are the same as in the decimal number,
but the decimal point in the product is shifted to the right as many places as there are zeros.

Let’s divide! Textbook Page No. 114

4 girls divided a 12 meter long ribbon among them. What length did each get?
It is not difficult to calculate this.
How about a 13 meter long ribbon?
12 meter divided into 4 equal parts give 3 meter long pieces; the remaining 1 meter divided into 4 gives $$\frac{1}{4}$$ meter. Altogether 3$$\frac{1}{4}$$ meters.

So, each gets 3$$\frac{1}{4}$$ meters
We can write this as 13 ÷ 4 = 3$$\frac{1}{4}$$
We can also write it as a decimal.
$$\frac{1}{4}$$ meter means 25 centimeter; that is, 0.25 meters.
So, instead of 3$$\frac{1}{4}$$ metrer, we can write 3.25 meters.

Look at this problem;

A square is made with a 24.8 centimeter long rope. What is the length of its side?
To find the length of a side, 24.8 must be divided into four equal parts.
24.8 centimeters means 24 centimeters and 8 millimeters.
24 centimeters divided into four equal parts give 6 centimeters each.
The remaining 8 millimeters divided into four equal parts give 2 millimeters each.
Thus the length of a side is 6 centimeters and 4 millimeters, that is 6.2 centimeters.
This problem also we can write using numbers only.
24.8 ÷ 4
The way we found the answer can also be written using just numbers.

24.8 mean 24 and 8 tenths. Dividing each by 4 gives 6 and 2 tenths; that is 6.2
These operations can be written in short hand as shown on the right.

A line of length 13.2 centimeters is divided into 3 equal parts .What is the length of each part?
We first divide 12 centimeters of 13.2 centimeters into 3 equal parts, getting 4 centimeter long parts; 1 centimeter and 2 millimeters remaining.
That is, 12 millimeters are left.

Dividing this into 3 equal parts gives 4 millimeters each. So, 13.2 centimeters divided into 3 equal parts give 4 centimeters and 4 millimeters as the length of a part.
That is 4.4 centimeters.
How about writing this as a division of numbers?
13.2 ÷ 3 = 4.4

How did we do this?
13.2 mean 13 and 2 tenths. in this, dividing 13 by 3 gives quotient 4 and remainder 1.

Changing this 1 to tenths and adding them to the 2 tenths already there, we get 12 tenths. 12 divided by 3 gives 4.
Thus we get 4 and 4 tenths; that is 4.4.
These operations also we can write in shorthand.

Let’s look at another problem:

4 people shared 16.28 kilograms of rice. How much does each get?
If 16 kilograms is divided in to 4 equal parts, how much is each part?
0.28 kilograms means 280 grams.
What if we divide 280 grams into 4?
So, how much does each get?
How about writing this using only numbers?
16.28 ÷ 4 = 4.07
16.28 means 16 and 2 tenths and 8 hundredths.
16 divided by 4 gives 4.

Changing 2 tenths to 20 hundredths and adding to the original 8 hundredths give 28 hundredths.
28 divided by 4 gives 7
So the total quotient is 4 and 7 hundredths.
That is 4.07.
The operation can be written like this:

25.5 kilograms of sugar is packed into 6 bags of the same size. How much is in each bag?
24 kilograms divided into 6 equal parts give 4 kilograms each. The remaining 1.5 kilograms, changed to grams are 1500 grams.
Dividing this into 6 equal parts gives 1500 ÷ 6 = 250 grams.

So one bag contains 4 kilograms and 250 grams; that is 4.250 kilograms.
We usually write this as 4.25 kilograms.
As numbers, we find
25.5 ÷ 6 = 4.25
The method of finding the answer can also be written using only numbers.
25.5 means 25 and 5 tenths.
25 divided by 6 gives 4 and remainder 1.
The remaining 1, changed to tenths and added to the original 5 tenths give 15 tenths; divided this by 6 gives 2 tenths and remainder 3 tenths.

These 3 tenths can be changed into 30 hundredths ; and this divided by 6 gives 5 hundredths.
What then is the total quotient?
4 and 2 tenths and 5 hundredths
That is ,4.25
Let’s write these operations in shorthand.

Textbook Page No. 118

Question 1.
The total amount of milk given to the children in a school for the 5 days of last week is 132.575 liters. How much was given on average each day?
26.515 liters
Explanation:
Total milk given in last 5 days = 132.575 liters
Number of days = 5
Average milk given on each day = Total milk given by number of days.
= 132.575 ÷ 5
= 26.515 liters

Question 2.
8 people shared 33.6 kilograms of rice. Sujitha divided her share into three equal parts and gave one part to Razia. How much did Razia get?
1.4 kg
Explanation:
Number of people = 8
Total rice shared = 33.6 kilograms.
No. of kgs each person got = 33.6 ÷ 8 = 4.2 kg
Sujitha divided her share into three equal parts = 4.2 ÷ 3 = 1.4 kg
Razia gets 1.4 kg share of Sujitha.

Question 3.
A ribbon of length 0.8 meters is divided into 16 equal parts. What is the length of each part’?
Explanation:
A ribbon of length 0.8 meters is divided into 16 equal parts.
1 m = 100 cm
0.8 m = 100 x 0.8 = 80 cm
The length of each part = 80 ÷ 16 = 5 cm

Question 4.
Do the problems below:
i) 54.5 ÷ 5
10.9
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern.

ii) 14.24 ÷ 8
1.78
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

iii) 56.87 ÷ 11
5.17
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

iv) 3.1 ÷ 2
1.55
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

v) 35.523 ÷ 3
11.841
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

vi) 36.48 ÷ 12
3.4
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a patter

vii) 16.56 ÷ 9
1.84
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

viii) 32.454 ÷ 4
8.1135
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

ix) 425.75 ÷ 25
17.03
Explanation:
Place the decimal point in the quotient directly above the decimal point in the dividend.
Divide the same way you would divide with whole numbers.
Divide until there is no remainder, or until the quotient begins to repeat in a pattern

Question 5.
Given 105.728 ÷ 7 = 15.104, find the answer to the problems below, with out actual division.
i) 1057.28 ÷ 7
151.04
Explanation:
1057.28 mean 1057 and 2 tenths, 8 hundredths.
in this, dividing 1057 by 7 gives quotient 151.
Changing this 2 tenths to 20 hundredths and adding to the original 8 hundredths gives 28 hundredths.
Then, we get 28 ÷ 7 = 4
Thus we get 151 and 2 tenths and hundredths; that is 151.04.

ii) 1.05728 ÷ 7
0.15104
Explanation:
1.05728 mean 1 and 0 tenths, 5 hundredths, 7 thousandths, 2 ten thousandths and 8 lakhs.
count the number of decimals and move the decimals from right in the quotient.
we get 1.057281 ÷ 7 0.15104

Question 6.
A number multiplied by 9 gives 145.71.  What is the number?
16.19
Explanation:
Let the number be x.
9x = 145.71
x = $$\frac{145.71}{9}$$
= $$\frac{145.71 × 100}{9 × 100}$$
= $$\frac{14571}{900}$$
= 16.19
So, 16.19 x 9 = 145.71

16.34 ÷ 10 = 163.4
25.765 ÷ 100 = _____.
347.5 ÷ 100 = ______.
238.4 ÷ 1000 = _____.
What have you found out about dividing a number in decimal form by 10, 100, 1000 and so on?
When we divide a decimal by 10, 100 and 1000,
the place value of the digits decreases.
The digits move to the right since the number gets smaller,
but the decimal point does not move.
Explanation:
When we observe the below division, there is no change in decimal places.
16.34 ÷ 10 = 163.4
25.765 ÷ 100 = 0.25765
347.5 ÷ 100 = 3.475
238.4 ÷ 1000 = 0.2384

Other Divisions

A rope of length 8.4 meters is cut into 0.4 meter long pieces. How many pieces can we make?
8.4 meters is 840 centimeters and 0.4 meter is 40 centimeters. So the number of pieces is 840 ÷ 40 = 21
We can write this as
8.4 ÷ 0.4 = 21
What does this mean?
8.4 is 21times 0.4
How about doing this with fractions?
84 = $$\frac{84}{10}$$, 0.4 = $$\frac{4}{10}$$
$$\frac{84}{10}$$ ÷ $$\frac{4}{10}$$ means, finding out the number, $$\frac{4}{10}$$ of which is $$\frac{84}{10}$$.

And we know that it is $$\frac{10}{4}$$ times $$\frac{84}{10}$$.
That is $$\frac{84}{10}$$ ÷ $$\frac{4}{10}$$ = $$\frac{84}{10}$$ × $$\frac{10}{4}$$ = 21
That is, $$\frac{84}{10}$$ ÷ $$\frac{4}{10}$$ = $$\frac{84}{10}$$ ÷ $$\frac{10}{4}$$ = 21
Can we compute 36.75 ÷ 0.5 like this?
36.75 = $$\frac{3675}{100}$$, 0.5 = $$\frac{5}{10}$$
$$\frac{3675}{100}$$ ÷ $$\frac{5}{10}$$ = $$\frac{3675}{100}$$ × $$\frac{10}{5}$$ = $$\frac{735}{10}$$
That is, 36.75 ÷ 0.5 = 73.5
We can also write $$\frac{36.75}{0.5}$$ = 73.5
So how do we find $$\frac{48.72}{0.12}$$?
$$\frac{48.72}{0.12}$$ = 48.72 ÷ 0.12 = $$\frac{4872}{100}$$ ÷ $$\frac{12}{100}$$
= $$\frac{4872}{100}$$ × $$\frac{100}{12}$$
= $$\frac{4872}{12}$$
= 406

Textbook Page No. 119

Question 1.
The area of a rectangle is 3.25 square meters and its length is 2.5 centimeters. What is its breadth’?
1.3 meters
Explanation:
Area of rectangle = length x breadth
The area of a rectangle is 3.25 square meters,
length is 2.5 centimeters.
breadth = $$\frac{3.25}{2.5}$$
b = 1.3 meters

Question 2.
A can contains 4.05 liters of coconut oil. It must be filled in to 0.45 liter bottles. How many bottles are needed?
9 bottles.
Explanation:
A can contains 4.05 liters of coconut oil.
Capacity of one bottle = 0.45 liters
Number of bottles required = $$\frac{4.05}{0.45}$$
Divided the numerator and denominator by 100
= $$\frac{405}{45}$$ = $$\frac{81}{9}$$
= 9 bottles.

Question 3.
Calculate the quotients below:

i) $$\frac{35.37}{0.03}$$
1,179
Explanation:
$$\frac{35.37}{0.03}$$ = 35.37 ÷ 0.03
= $$\frac{3537}{100}$$ ÷ $$\frac{3}{100}$$
= $$\frac{3537}{100}$$ × $$\frac{100}{3}$$
= $$\frac{3537}{3}$$
= 1179

ii) $$\frac{10.92}{2.1}$$
52
Explanation:
$$\frac{10.92}{2.1}$$ = 10.92 ÷ 2.1
= $$\frac{1092}{100}$$ ÷ $$\frac{21}{10}$$
= $$\frac{1092}{100}$$ × $$\frac{10}{21}$$
= $$\frac{10920}{210}$$
= 52

iii) $$\frac{40.48}{1.1}$$
3,680
Explanation:
$$\frac{40.48}{1.1}$$ = 4048 ÷ 11
= $$\frac{4048}{100}$$ ÷ $$\frac{11}{10}$$
= $$\frac{4048}{100}$$ × $$\frac{10}{11}$$
= $$\frac{40480}{11}$$
= 3680

iv) $$\frac{0.045}{0.05}$$
0.9
Explanation:
$$\frac{0.045}{0.05}$$ = 0.045 ÷ 0.05
= $$\frac{45}{1000}$$ ÷ $$\frac{5}{100}$$
= $$\frac{45}{1000}$$ × $$\frac{100}{5}$$
= $$\frac{45}{50}$$
= 0.9

v) 0.001 ÷ 0.1
0.01
Explanation:
0.001 ÷ 0.1
= $$\frac{1}{1000}$$ ÷ $$\frac{1}{10}$$
= $$\frac{1}{1000}$$ × $$\frac{10}{1}$$
= $$\frac{1}{100}$$
= 0.01

vi) 5.356 ÷ 0.13
41.2
Explanation:
5.356 ÷ 0.13
= $$\frac{5356}{1000}$$ ÷ $$\frac{13}{100}$$
= $$\frac{5356}{1000}$$ × $$\frac{100}{13}$$
= $$\frac{5356}{130}$$
= 41.2

vii) $$\frac{0.2 \times 0.4}{0.02}$$
4
Explanation:
$$\frac{0.2 \times 0.4}{0.02}$$
$$\frac{0.08}{0.02}$$ = 4

viii) $$\frac{0.01 \times 0.01}{0.001 \times 0.1}$$
1
Explanation:
$$\frac{0.01 \times 0.01}{0.001 \times 0.1}$$
$$\frac{0.0001}{0.0001}$$ = 1

Question 4.
12125 divided by which number gives 1.2125?
10,000
Explanation:
Let number be divided by x.
$$\frac{12125}{x}$$ = 1.2125
x = $$\frac{12125}{1.2125}$$
x = 10,000
12125 divided by 10,000 gives 1.212

Question 5.
0.01 multiplied by which number gives 0.00001?
0.001
Explanation:
Let the number be multiplied by x.
0.01x = 0.00001
x = $$\frac{0.00001}{0.01}$$
multiply both numerator and denominator with 100
x = $$\frac{0.00001}{0.01}$$ x $$\frac{100}{100}$$
x = 0.001
0.01 x 0.001 = 0.00001

Fractions and decimals

Fractions written as decimals are of denominators 10,100, 1000 and so on.
For some fractions, we can first change the denominator into one of these and then write in decimal form. For example,
$$\frac{1}{2}$$ = $$\frac{5}{10}$$ = 0.5
$$\frac{1}{4}$$ = $$\frac{25}{100}$$ = 0.25
$$\frac{3}{4}$$ = $$\frac{75}{100}$$ = 0.75

How do we write $$\frac{1}{8}$$ in decimal form?
8 = 2 × 2 × 2
So, multiplying 8 by three 5’s we can make it a product of 10’s.
8 × (5 × 5 × 5) = (2 × 2 × 2) × (5 × 5 × 5)
= (2 × 5) × (2 × 5) × (2 × 5)
= 10 × 10 × 10 = 1000
5 × 5 × 5 = 125, right? So
$$\frac{1}{8}$$ = $$\frac{125}{8 \times 125}$$ = $$\frac{125}{1000}$$ = 0.125
In much the same way,
$$\frac{5}{8}$$ = $$\frac{5 \times 125}{8 \times 125}$$ = $$\frac{625}{1000}$$ = 0.625

How about $$\frac{1}{40}$$ ?
40 = (2 × 2 × 2) × 5
To get a product of 10’s we have to multiply 40 by two 5’s; that is
40 × 25 = (2 × 2 × 2 × 5) × (5 × 5)
= (2 × 5) × (2 × 5) × (2 × 5)
= 10 × 10 × 10
= 1000
So,
$$\frac{1}{40}$$ = $$\frac{25}{40 \times 25}$$ = $$\frac{25}{1000}$$ = 0.025
And $$\frac{21}{40}$$?
$$\frac{21}{40}$$ = $$\frac{21 \times 25}{40 \times 25}$$ = $$\frac{525}{1000}$$ = 0.525

Similarly, since 125 × 8 = 1000, we can write
$$\frac{121}{125}$$ = $$\frac{121 \times 8}{125 \times 8}$$ = $$\frac{968}{1000}$$ = 0.968
Thus we can find the decimal form of any fraction whose denominator is a multiple of 2’s and 5’s.

Now look at this problem:
24 kilograms of sugar are packed into 25 packets of the same size. How much does each packet contain?
24 kilograms means 24000 grams. So each packet contains $$\frac{24000}{25}$$ grams.
$$\frac{24000}{25}$$ = 960

Thus each packet contains 960 grams or 0.96 kilograms.
We can do this in a different way. Each packet contains $$\frac{24}{25}$$ kilograms.
$$\frac{24}{25}$$ = $$\frac{24 \times 4}{25 \times 4}$$ = $$\frac{96}{100}$$ = 0.96
So, one packet contains 0.96 kilograms.

Textbook Page No. 121

Question 1.
Find the decimal forms of the fractions below:

i) $$\frac{3}{5}$$
0.6
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.
$$\frac{3}{5}$$ = 0.6

ii) $$\frac{7}{8}$$
0.875
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.

iii) $$\frac{5}{16}$$
0.3125
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.

iv) $$\frac{3}{40}$$
0.075
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.

v) $$\frac{3}{32}$$
0.09375
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.

vi) $$\frac{61}{125}$$
0.488
Explanation:
To find the decimal of a fraction,
divide the numerator by the denominator.
Because the number in the numerator is smaller than the number in the denominator,
place the decimal point after it and add zeros.

Question 2.
Write the answer to the questions below in decimal form.
(i) 3 liters of milk is used to fill 8 identical bottles. How much does each bottle contain?
0.375 liters.
Explanation:
3 liters of milk is used to fill 8 identical bottles.
Each bottle contains = $$\frac{3}{8}$$
= 0.375 liters.

(ii) A 17 meter long string is cut into 25 equal parts. What is the length of each part?
0.68 meters
Explanation:
A 17 meter long string is cut into 25 equal parts.
The length of each part = $$\frac{17}{25}$$
= 0.68 meters

(iii) 19 kilograms of rice is divided among 20 people. How much does each get?
0.95 kilograms.
Explanation:
19 kilograms of rice is divided among 20 people.
Total kilograms of rice each get = $$\frac{19}{20}$$
= 0.95 kg

Question 3.
What is the decimal form of $$\frac{1}{2}$$ + $$\frac{1}{4}$$ + $$\frac{1}{8}$$ + $$\frac{1}{16}$$?
0.9375
Explanation:
$$\frac{1}{2}$$ + $$\frac{1}{4}$$ + $$\frac{1}{8}$$ + $$\frac{1}{16}$$
numerators are same denominators are different, so find the LCM of denominators
= $$\frac{(1 ×16) + (1 × 8) + (1 × 4) + (1× 2)}{32}$$
= $$\frac{16 + 8 + 4 + 2}{32}$$
= $$\frac{30}{32}$$
= 0.9375

Question 4.
A two digit number divided by another two digit number gives 4.375.What are the numbers’?
The two digit numbers are 70 and 16.
Explanation:
Given number is 4.375
convert the decimal number to whole number,
$$\frac{4375}{1000}$$
find the factors of above fraction,
factors of 4375 = 5 x 5 x 5 x 5 x 7
factors of 1000 = 2 x 2 x 2 x 5 x 5 x 5
Divide the common factors in both the numbers,
$$\frac{35}{8}$$
multiply both numerator and denominator with 2,
$$\frac{35 × 2}{8 × 2}$$ = $$\frac{70}{16}$$ = 4.375

Question 1.
What is the volume of a rectangular block of length 25.5 centimeters, breadth 20.4 centimeters and height 10.8 centimeters?
5618.16 cm3
Explanation:
Given, length 25.5 centimeters, breadth 20.4 centimeters and height 10.8 centimeters.
Volume of a cuboid = l x b x h
V = 25.5 x 20.4 x 10.8 = 5618.16 cm3

Question 2.
The heights of three boys sitting on a bench are 130.5 centimeters 128.7 centimeters and 134.6 centimeters .What is the average height’?
131.26 centimeters.
Explanation:
The heights of three boys sitting on a bench are,
130.5 centimeters 128.7 centimeters and 134.6 centimeters.
The average height of 3 boys = $$\frac{130.5 + 128.7 + 134.6}{3}$$
= 393.8 ÷ 3 = 131.26

Question 3.
Calculate $$\frac{4 \times 3.06}{3}$$.
4.08
Explanation:
$$\frac{4 \times 3.06}{3}$$
$$\frac{12.24}{3}$$ = 4.08

Question 4.
The price of 22 pencils is 79.20 rupees. What is the price of 10 pencils’?
36 rupees
Explanation:
The price of 22 pencils is 79.20 rupees.
Price of each pencil = 79.20 ÷ 22 = 3.60 rupees
The price of 10 pencils = 3.60 x 10 = 36 rupees.

Question 5.
Calculate the following:

i) $$\frac{2.3 \times 3.2}{0.4}$$
18.4
Explanation:
$$\frac{2.3 \times 3.2}{0.4}$$
$$\frac{7.36}{0.4}$$ = 18.4

ii) $$\frac{0.01 \times 0.001}{0.1 \times 0.01}$$
0.01
Explanation:
$$\frac{0.01 \times 0.001}{0.1 \times 0.01}$$
$$\frac{0.00001}{0.001}$$ = 0.01

Question 6.
Dividing 0.1 by which number gives 0.001?
x = $$\frac{0.1}{0.001}$$
x = $$\frac{100}{1}$$