2. Write the length of Ravi’s and Raju’s pencil in ‘cm’ using decimals.

      3. Make three more examples similar to the one given in question 1 and solve them.

Representing Decimals on number line

We represented fractions on a number line. Let us now represent decimals too on a number line. Let us represent 0.6 on a number line.
We know that 0.6 is more than zero but less than one. There are 6 tenths in it. Divide the unit length between 0 and 1 into 10 equal parts and take 6 parts as shown below :

  

Write five numbers between 0 and 1 and show them on the number line.
Can you now represent 2.3 on a number line? Check, how many ones and tenths are there in 2.3. Where will it lie on the number line? Show 1.4 on the number line.

Example 1

Write the following numbers in the place value table : (a) 20.5 (b) 4.2

Solution

Let us make a common place value table, assigning appropriate place value to the digits in the given numbers. We have,

   

Example 2

Write each of the following as decimals : (a) Two ones and five-tenths (b) Thirty and one-tenth

Solution

(a) Two ones and five-tenths = 2 + \(\frac{5}{10}\) = 2.5
(b) Thirty and one-tenth = 30 + \(\frac{1}{10}\) = 30.1

Example 3

Write each of the following as decimals :
(a) 30 + 6 + \(\frac{2}{10}\)       (b) 600 + 2 + \(\frac{8}{10}\)

Solution

(a) 30 + 6 + \(\frac{2}{10}\)

How many tens, ones and tenths are there in this number? We have 3 tens, 6 ones and 2 tenths.
Therefore, the decimal representation is 36.2.
(b) 600 + 2 + \(\frac{8}{10}\)
Note that it has 6 hundreds, no tens, 2 ones and 8 tenths.
Therefore, the decimal representation is 602.8

Fractions as decimals

We have already seen how a fraction with denominator 10 can be represented using decimals.
Let us now try to find decimal representation of (a) \(\frac{11}{5}\) (b) \(\frac{1}{2}\)

We know that \(\frac{{11}}{5} = \frac{{22}}{{10}} = \frac{{20 + 2}}{{10}} = \frac{{20}}{{10}} + \frac{2}{{10}} = 2 + \frac{2}{{10}} = 2.2\)

Therefore, \(\frac{22}{10}\) = 2.2 (in decimal notation.)
(b) In \(\frac{1}{2}\) , the denominator is 2. For writing in decimal notation, the denominator should be 10. We already know how to make an equivalent fraction. So, \(\frac{1}{2} = \frac{{1 \times 5}}{{2 \times 5}} = \frac{5}{{10}} = 0.5\)

Try These

• Write \(\frac{3}{2},\frac{4}{5},\frac{8}{5}\) in decimal notation

         

 

       

 

2. Write the following decimals in the place value table.
(a) 19.4    (b) 0.3    (c) 10.6    (d) 205.9

3. Write each of the following as decimals :

• Seven-tenths = 
• Two tens and nine-tenths = 
• Fourteen point six = 
• One hundred and two ones = 
• Six hundred point eight = 

8. Show the following numbers on the number line.
(a) 0.2    (b) 1.9    (c) 1.1    (d) 2.5

9. Write the decimal number represented by the points A, B, C, D on the given number line

         

A =
B =
C =
D =

10. a) The length of Ramesh’s notebook is 9 cm 5 mm. What will be its length in cm? 

     b) The length of a young gram plant is 65 mm. Express its length in cm. 

8.3 Hundredths

David was measuring the length of his room. He found that the length of his room is 4 m and 25 cm.
He wanted to write the length in metres.
Can you help him? What part of a metre will be one centimetre?

      

1 cm = \(\frac{1}{100}\) m or one-hundredth of a metre.
This means 25 cm = \(\frac{25}{100}\) m.
Now \(\frac{1}{100}\) means 1 part out of 100 parts of a whole. As we have done for \(\frac{1}{10}\) , let us try to show this pictorially.
Take a square and divide it into ten equal parts.
What part is the shaded rectangle of this square?
It is \(\frac{1}{10}\) or one-tenth or 0.1, see Fig (i).

    

Now divide each such rectangle into ten equal parts.
We get 100 small squares as shown in Fig (ii).
Then what fraction is each small square of the whole square?

     

Each small square is \(\frac{1}{100}\) or one-hundredth of the whole square. In decimal notation, we write \(\frac{1}{100}\) = 0.01 and read it as zero point zero one.
What part of the whole square is the shaded portion, if we shade 8 squares, 15 squares, 50 squares, 92 squares of the whole square? Take the help of following figures to answer.

   

    

Let us look at some more place value tables.

   

The number shown in the table above is \(2 + \frac{4}{{10}} + \frac{3}{{100}}\).In decimals, it is written as 2.43, which is read as ‘two point four three’.

Example 4

Fill the blanks in the table using ‘block’ information given below and write the corresponding number in decimal form.

   

Solution

  

The number is \(100 + 30 + 2 + \frac{1}{{10}} + \frac{5}{{100}} = 132.15\)

Example 5

Fill the blank in the table and write the corresponding number in decimal form using ‘block’ information given below.

    

     

     Solution 

     

    

Therefore, the number is 1.42.

Example 6

Given the place value table, write the number in decimal form.

  

  Solution

  The number is \(2 \times 100 + 4 \times 10 + 3 \times 1 + 2 \times \frac{1}{{10}} + 5 \times \left( {\frac{1}{{100}}} \right)\)

                 \( = 200 + 40 + 3 + \frac{1}{{10}} + \frac{5}{{100}} = 243.25\)

The first digit 2 is multiplied by 100; the next digit 4 is multiplied by 10 i.e. ( \(\frac{1}{10}\) of 100); the next digit 3 is multiplied by 1. After this, the next multiplying factor is \(\frac{1}{10}\) ; and then it is \(\frac{1}{100}\) i.e. ( \(\frac{1}{10}\) of \(\frac{1}{10}\) ).
The decimal point comes between ones place and tenths place in a decimal number. It is now natural to extend the place value table further, from hundredths to \(\frac{1}{10}\) of hundredths i.e. thousandths.
Let us solve some examples.

Example 7

Write as decimals. (a) \(\frac{4}{5}\) (b) \(\frac{3}{4}\) (c) \(\frac{7}{1000}\) 

Solution

• We have to find a fraction equivalent to whose denominator is 10. \(\frac{4}{5} = \frac{{4 \times 2}}{{5 \times 2}} = \frac{8}{{10}} = 0.8\)
• Here, we have to find a fraction equivalent to \(\frac{3}{4}\) with denominator 10 or 100. There is no whole number that gives 10 on multiplying by 4, therefore, we make the denominator 100 and we have, \(\frac{3}{4} = \frac{{3 \times 25}}{{4 \times 25}} = \frac{{75}}{{100}} = 0.75\)
• Here, since the tenth and the hundredth place is zero.
  Therefore, we write \(\frac{7}{1000}\) = 0.007

Example 8

Write as fractions in lowest terms.
(a) 0.04    (b) 2.34    (c) 0.342

Solution

a) \(0.04 = \frac{4}{{100}} = \frac{1}{{25}}\)

b) \(2.34 = 2 + \frac{{34}}{{100}} = 2 + \frac{{34 \div 2}}{{100 \div 2}} = 2 + \frac{{17}}{{50}} = 2\frac{{17}}{{50}}\)

c) \(0.342 = \frac{{342}}{{1000}} = \frac{{342 \div 2}}{{1000 \div 2}} = \frac{{171}}{{500}}\)

Example 9

Write each of the following as a decimal.

a) \(200 + 30 + 5 + \frac{2}{{10}} + \frac{9}{{100}}\)

b) \(50 + \frac{1}{{10}} + \frac{6}{{100}}\)

c) \(16 + \frac{3}{{10}} + \frac{5}{{1000}} \)

Solution 

a) \(200 + 30 + 5 + \frac{2}{{10}} + \frac{9}{{100}} = 235 + 2 \times \frac{1}{{10}} + 9 \times \frac{1}{{100}} = 235.29\)

b) \(50 + \frac{1}{{10}} + \frac{6}{{100}} = 50 + 1 \times \frac{1}{{10}} + 6 \times \frac{1}{{100}} = 50.16\)

c) \(16 + \frac{3}{{10}} + \frac{5}{{1000}} = 16 + \frac{3}{{10}} + \frac{0}{{100}} + \frac{5}{{1000}}\)

     \( = 16 + 3 \times \frac{1}{{10}} + 0 \times \frac{1}{{100}} + 5 \times \frac{1}{{1000}} = 16.305\)

 

Example 10

Write each of the following as a decimal.

• Three hundred six and seven-hundredths
• Eleven point two three five
• Nine and twenty five thousandths

b. Eleven point two three five = 11.235

c. Nine and twenty five thousandths \( = 9 + \frac{{25}}{{1000}} = 9 + \frac{0}{{10}} + \frac{2}{{100}} + \frac{5}{{1000}} = 9.025\)

     since, 25 thousands  \( = \frac{{25}}{{1000}} = \frac{{20}}{{1000}} + \frac{5}{{1000}} = \frac{2}{{100}} + \frac{5}{{1000}}\)

    

E X E R C I S E 8.2

1. Complete the table with the help of these boxes and use decimals to write the number.

          

            

2. Write the numbers given in the following place value table in decimal form

      

3.    Write the following decimals in the place value table.

• 0.29
• 2.08
• 19.60
• 148.32
• 200.812

5. Write each of the following decimals in words.

• 0.03 = 
• 1.20 = 
• 108.56 = 
• 10.07 = 
• 0.032 = 
• 5.008 = 

  

  This means 0.1>0.07
Let us now compare the numbers 32.55 and 32.5. In this case , we first compare the whole part. We see that the whole part for both the nunbers is 32 and, hence, equal.
We, however, know that the two numbers are not equal. So, we now compare the tenth part. We find that for 32.55 and 32.5, the tenth part is also equal, then we compare the hundredth part.
We find,
32.55 = 32 + \(\frac{5}{10}\) + \(\frac{5}{100}\) and 32.5 = 32 + \(\frac{5}{10}\) + \(\frac{5}{100}\) , therefore, 32.55>32.5 as the hundredth part of 32.55 is more.

Example 11

Which is greater?
(a) 1 or 0.99     (b) 1.09 or 1.093

Solution

a) \(1 = 1 + \frac{0}{{10}} + \frac{0}{{100}};0.99 = 0 + \frac{9}{{10}} + \frac{9}{{100}}\)

The whole part of 1 is greater than that of 0.99.
Therefore, 1 > 0.99

b) \(1.09 = 1 + \frac{0}{{10}} + \frac{9}{{100}} + \frac{0}{{1000}};1.093 = 1 + \frac{0}{{10}} + \frac{9}{{100}} + \frac{3}{{1000}}\)

In this case, the two numbers have same parts upto hundredth.
But the thousandths part of 1.093 is greater than that of 1.09.
Therefore, 1.093 > 1.09.

E X E R C I S E 8.3

1. Which is greater?
   • 0.3 or 0.4 = 
   • 0.07 or 0.02 = 
   • 3 or 0.8 = 
   • 0.5 or 0.05 = 
   • 1.23 or 1.2 = 
   • 0.099 or 0.19 = 
   • 1.5 or 1.50 = 
   • 1.431 or 1.490 = 
   • 3.3 or 3.300 = 
   • 5.64 or 5.60 = 
2. Make five more examples and find the greater number from them.

8.5 Using Decimals

8.5.1 Money

We know that 100 paise = Re 1
Therefore, 1 paise = Re \(\frac{1}{100}\) = Re 0.01
So, 65 paise = Re \(\frac{65}{100}\) = Re 0.65
and 5 paise = Re \(\frac{5}{100}\) = Re 0.05
What is 105 paise? It is Re 1 and 5 paise = Rs 1.05

Try These

• Write 2 rupees 5 paise and 2 rupees 50 paise in decimals? = 
• Write 20 rupees 7 paise and 21 rupees 75 paise in decimals? = 

 

8.5.2 Length

Mahesh wanted to measure the length of his table top in metres. He had a 50 cm scale. He found that the length of the table top was 156 cm. What will be its length in metres?

   

Mahesh knew that
1 cm = \(\frac{1}{100}\) m or 0.01 m
Therefore, 56 cm = \(\frac{56}{100}\) m = 0.56 m
Thus, the length of the table top is
156 cm = 100 cm + 56 cm
= 1 m + \(\frac{56}{100}\) m = 1.56 m.
Mahesh also wants to represent this length pictorially. He took squared papers of equal size and divided them into 100 equal parts.
He considered each small square as one cm.

    

Try These

• Can you write 4 mm in ‘cm’ using decimals? = 
• How will you write 7cm 5 mm in ‘cm’ using decimals? = 
• Can you now write 52 m as ‘km’ using decimals? How will you write 340 m as ‘km’ using decimals? How will you write 2008 m in ‘km’? = 

E X E R C I S E 8.4

1. Express as rupees using decimals.
   • 5 paise = 
   • 75 paise = 
   • 20 paise = 
   • 50 rupees = 
   • 90 paise = 
   • 725 paise = 
2. Express as metres using decimals.
   • 15 cm = 
   • 6 cm = 
   • 2 m 45 cm = 
   • 9 m 7 cm = 
   • 419 cm = 
3. Express as cm using decimals.
   • 5 mm = 
   • 60 mm = 
   • 164 mm = 
   • 9 cm 8 mm = 
   • 419 cm = 
4. Express as km using decimals.
   • 8 m = 
   • 88 m = 
   • 8888 m = 
   • 70 km 5 m = 
5. Express as kg using decimals.
   • 2 g = 
   • 100 g = 
   • 3750 g = 
   • 5 kg 8 g = 
   • 26 kg 50 g = 

8.6 Addition of Numbers with Decimals

Do This

Add 0.35 and 0.42.
Take a square and divide it into 100 equal parts.
Mark 0.35 in this square by shading 3 tenths and colouring 5 hundredths.
Mark 0.42 in this square by shading 4 tenths and colouring 2 hundredths.
Now count the total number of tenths in the square and the total number of hundredths in the square.

    

Therefore, 0.35 + 0.42 = 0.77
Thus, we can add decimals in the same way as whole numbers.
Can you now add 0.68 and 0.54?

   

Thus, 0.68 + 0.54 = 1.22

Try These

• 0.29 + 0.36 = 
• 0.7 + 0.08 = 
• 1.54 + 1.80 = 
• 2.66 + 1.8 =