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

## 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 = $$\frac{100}{1}$$