Chapter 6.1 - Place values in Decimals

In the previous section we saw the relation between:
• the 'position' of a digit after the decimal point and
• the 'power' of 10 in the denominator of it's fraction form
In this section we will see a pictorial representation of that relation:

We have seen that 0.36 = ³⁶⁄₁₀₀ = ³⁄₁₀  + ⁶⁄₁₀₀
 ³⁄₁₀ is 3 parts taken out of 10 equal parts. This can be shown pictorially as in fig.6.6(a):

Fig.6.6

In fig.(a), the whole is divided into 10 equal parts by the horizontal lines. From those 10, 3 parts (shown in green) are taken out.  But we need 6 more. This 6 cannot be taken from the divisions in fig.(a) because:
• The smallest division here is 10. 
• Our requirement 6 is smaller than the smallest division 10. 
So to take out 6, we have to divide the whole into still smaller parts. So we divide the whole into 100 equal parts. This is shown in fig.(b). The horizontal and vertical lines together divide the whole into 100 equal parts. Among those 100, 6 nos. are marked in magenta color. Those 6 form 6/100 of the whole.

So to get 0.36, we have to add 3 from the 10 equal parts and 6 from the 100 equal parts. That is.,
0.36 = ³⁄₁₀  + ⁶⁄₁₀₀ 
• 3 is having the 1^st power of 10 in the denominator and
• 6 is having the 2^nd power of 10 in the denominator

Thus we have a pictorial representation of the relation. Based on this relation, we can give accurate 'place value names' for each of the digits coming after the decimal point:

■ 3, which is the 1^st digit coming just after the decimal point is obtained from dividing the whole into 10 equal parts. So the place value of the 1^st digit is called tenths.
■ 6, which is the 2^nd digit coming after the decimal point is obtained from dividing the whole into 100 equal parts. So the place value of the 2^nd digit is called hundredths


This is shown in the fig.6.7 below:

Fig.6.7

In the same way, the 3^rd position after the decimal point is called thousandths, and so on.

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We will now see some solved examples based on the above discussion:
Solved example 6.1
Write the following fractions in decimal form: (i) 2/5  (ii) 6/25  (iii) 1/8  (iv) 5/8  (v) 3/20
Solution:
(i) • We have to convert ²⁄₅ into an equivalent fraction which has a denominator 10, or 100, or 1000 ... so on, which ever is suitable. Let us try 10:
• ²⁄₅ = ^x⁄₁₀. It is clear that if we multiply the denominator 5 by 2, we will get 10. So the numerator must also be multiplied by 2. We will get x = 2 × 2  = 4
• So the equivalent fraction is calculated as:  ²⁄₅  =   ^{(2 × 2)}⁄_{(5 × 2)}  =  ⁴⁄₁₀   
• Thus the decimal form is 0.4

(ii) • We have to convert ⁶⁄₂₅ into an equivalent fraction which has a denominator 10, or 100, or 1000 ... so on, which ever is suitable. Let us try 10:
• ⁶⁄₂₅ = ^x⁄₁₀. There is no whole number which when multiplied with 25 will give 10. So we will try 100:
• ⁶⁄₂₅ = ^x⁄₁₀₀. It is clear that if we multiply the denominator 25 by 4, we will get 100. So the numerator must also be multiplied by 4. We will get x = 6 × 4  = 24
• So the equivalent fraction is calculated as:  ⁶⁄₂₅  =   ^{(6 × 4)}⁄_{(25 × 4)}  =  ²⁴⁄₁₀₀   
• Thus the decimal form is 0.24

(iii) • We have to convert ¹⁄₈ into an equivalent fraction which has a denominator 10, or 100, or 1000 ... so on, which ever is suitable. Let us try 10:
• ¹⁄₈ = ^x⁄₁₀. There is no whole number which when multiplied with 8 will give 10. So we will try 100:
• ¹⁄₈ = ^x⁄₁₀₀. There is no whole number which when multiplied with 8 will give 100. So we will try 1000:
• ¹⁄₈ = ^x⁄₁₀₀₀. It is clear that if we multiply the denominator 8 by 125, we will get 1000. So the numerator must also be multiplied by 125. We will get x = 1 × 125  = 125
• So the equivalent fraction is calculated as:  ¹⁄₈  =   ^{(1 × 125)}⁄_{(8 × 125)}  =  ¹²⁵⁄₁₀₀₀   
• Thus the decimal form is 0.125

(iv) • We have to convert ⁵⁄₈ into an equivalent fraction which has a denominator 10, or 100, or 1000 ... so on, which ever is suitable. From previous example, we know that, for the denominator 8, 10 and 100 are not possible. We have to use 1000. We also know that 125 is the factor that has to be used for multiplication.
• So the equivalent fraction is calculated as:  ⁵⁄₈  =   ^{(5 × 125)}⁄_{(8 × 125)}  =  ⁶²⁵⁄₁₀₀₀   
• Thus the decimal form is 0.625

(v) • We have to convert ³⁄₂₀ into an equivalent fraction which has a denominator 10, or 100, or 1000 ... so on, which ever is suitable. Let us try 10:
• ³⁄₂₀ = ^x⁄₁₀. There is no whole number which when multiplied with 20 will give 10. So we will try 100:
• ³⁄₂₀ = ^x⁄₁₀₀. It is clear that if we multiply the denominator 20 by 5, we will get 100. So the numerator must also be multiplied by 5. We will get x = 3 × 5  = 15
• So the equivalent fraction is calculated as:  ³⁄₂₀  =   ^{(3 × 5)}⁄_{(20 × 5)}  =  ¹⁵⁄₁₀₀   
• Thus the decimal form is 0.15

Solved example 6.2

Write the place value of each of the digits in the following numbers:

(i) 0.85  (ii) 0.639  (iii) 0.079 (iv) 0.02  (v) 0.0208

Solution:

The place values are shown in the following table:

Solved example 6.3

Write the decimal represented by each of the following figs

Fig.6.8

Solution:

(a) • 8 parts are taken out of 10 equal parts + 4 parts are taken out of 100 equal parts

• So in fractional form it is ⁸⁄₁₀ + ⁴⁄₁₀₀

• From this we get the decimal form as 0.84 (8 in the tenths place and 4 in the hundredths place)

(b) • 5 parts are taken out of 10 equal parts + 9 parts are taken out of 100 equal parts

• So in fractional form it is ⁵⁄₁₀ + ⁹⁄₁₀₀

• From this we get the decimal form as 0.59 (5 in the tenths place and 9 in the hundredths place)

(c) • 3 parts are taken out of 10 equal parts + 8 parts are taken out of 100 equal parts

• So in fractional form it is ³⁄₁₀ + ⁸⁄₁₀₀

• From this we get the decimal form as 0.38 (3 in the tenths place and 8 in the hundredths place)

(d) • 7 parts are taken out of 10 equal parts + 2 parts are taken out of 100 equal parts

• So in fractional form it is ⁷⁄₁₀ + ²⁄₁₀₀

• From this we get the decimal form as 0.72 (7 in the tenths place and 2 in the hundredths place)

Solved example 6.4

An artist is trying to make a particular shade of yellow colour for his painting. He has several blocks (all of the same size) of 'perfect yellow' in his shelf. But he does not want perfect yellow. He wants a particular shade of yellow. For that, he must take 0.27 of a block. How would he obtain 0.27 of a block?

solution:

• The artist must take exact 0.27. Any other quantity would not give the required shade.

• We know that 0.27 = ²⁄₁₀ + ⁷⁄₁₀₀

• Given that all blocks are of equal size.

• So first he must take one block, divide it into 10 equal parts and take 2 parts from it

• Then he must take another block, divide it into 100 equal parts and take 7 parts from it

• The '2 out of 10', and the '7 out of 100' together will give 0.27 of a block. This is shown in the fig. below:

Fig.6.9

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In the next section we will discuss about comparison and addition of decimals

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