Decimal Fraction

A fraction, which has 10 or any power of 10 as its denominator is called a decimal fraction.
       eg, 1/10 = 0.1,    5/10 = 0.5,   1/100 = 0.01,   11/100 = 0.11,   4/1000 = 0.004, ......
    (a) The tenth part of a unit is denoted by 1/10 = 0.1 and read as point one or decimal one.
    (b) The hundredth part of a unit is denoted by 1/100 = 0.01 and read as point zero one or decimal zero one.
    (c) The thousandth part on unit is denoted by 1/1000 = 0.001 and read as point zero-zero one or decimal
        zero-zero one.
    (d) The number of figures, which follow the decimal point is the number of decimal point is the number of
        decimal places.
       eg, 12.405 has 3 decimal places in which 12 is the integral part and 405 is the decimal part.
       100.2222 has 4 decimal places in which 100 is the integral part 2222 is the decimal part.

To Convert a Decimal Fraction into a Vulgar Fraction (Generally, Prevalent Fraction)

To convert put '1' in the denomination under the decimal point and add as many zeroes as is the number of digits
after the decimal sign. Now, remove the decimal point and reduce the given fraction to its lowest terms.
eg,        0.15 = 15/100 = 3/20,  0.036 = 36/1000 = 9/250 
           5.25 = 5 + 0.25 = 5 + 25/100 
                                   = 5 + 1/4 
                                                   = 5(1/4) = 21/4 
                                        or,  5.25 = 525/100 = 21/4
(a) Adding zero to the right of decimal fraction does not change its value. eg, 0.4 = 0.40 = 0.400, etc.
(b) If the numerator and denominator of a fraction contains the same number of decimal places, then we can
    simply remove each of the decimal signs.
        eg,           4.55/8.16 = 455/816, 0.1/0.5 = 1/5, 0.003/0.007 = 3/7

Recurring Decimals

These are of two types

1. Pure Recurring Decimals

A decimal in which all the figures after the decimal point recur (occur again) is called a pure recurring 
decimal. eg, 2.‾5 is 2.55555...; 0.‾17 is o.17171717.....; 0.‾1 is 0.1111111............

Converting Pure Recurring Decimal into Vulgar Fraction

Write the repeated figures once only in the numerator and take as many nines in the denominator as the 
repeated figures.
           Example 1 : Express each of the following recurring decimals as vulgar fractions.
                                (i) 0.‾3   (ii) 0.‾12    (iii) 3.‾27 
           Solution. (i) 0.‾3 = 3/9 = 1/3
                    (ii) 0.‾12 = 12/99 = 4/33 
                    (iii)3.‾27 = 3 + 0.‾27 = 3 + 27/99 
                                               = 3 + 3/11 = 3 3/11 

2. Mixed Recurring Decimals

A decimal in which some figures do not repeat and some of them repeat, is called recurring decimal.
 eg, 0.45‾24 is 0.4524242424.....; 0.6‾23 is 0.6232323....; 3.1‾52 is 3.1525252......
 

Converting a Mixed Recurring Decimal into a Vulgar Fraction

In the numerator, take the difference between the number formed by all the digits after decimal point and 
that formed by non-repeating digits. In the denominator, take as many nines as there are repeating digits
and annex as many zeros as is the number of non-repeating digits.
          Example 2 Express each of the following mixed recurring decimal into vulgar fraction.
                      (i)0.3‾8      (ii) 0.45‾24      (iii) 7.42‾5
                  Solution. (i) 0.3‾8 = (38 - 3)/90 = 35/90 = 7/18 
                            (ii) 0.45‾24 = (4524 - 45)/9900 = 4479/9900
                                        (iii)7.42‾5 = 7 + 0.42‾5 = 7 + (425 - 42)/900 = 7 + 383/900 = 7 (383/900)
                  Example 3 Convert each of the following into vulgar fraction.
                        (i) 0.72     (ii) 5.06   (iii) 0.0095     (iv) 12.175
                  Solution. We have 
                            (i) 0.72 = 72/100 = 18/25 
                                        (ii)5.06 = 506/100 = 253/50 
                                        (iii) 0.0095 = 95/10000 = 19/2000
                                        (iv) 12.175 = 12175/1000 = 487/40 

Solved Examples

Type 1

To Find the Valueof (?)

Example 4 : 3 + 0.3 + 0.33 + 0.03 + 0.033 = ?
Solution.                        3.0
                                 0.3
                                 0.33
                                 0.03
                                 0.033
                                 -----
                                 3.693
                                 -----
        Hence,  ? = 3.693 
Example 5 : 0.789 + ? x 75 = 4.269
Solution. 7.5 x ? = 4.269 - 0.789 = 3.48
                         ? = 3.48/7.5 = 0.464
                                                 
Example 6 : 1.23 x ? - 3.6 ÷ 2.4 = 3.42 
Solution. 1.23 x ? - 3.6/2.4 = 3.42 => 1.23 x ? - 1.5 = 3.42 
                 ? = (3.42 + 1.5)/1.23 = 4.92/1.23  = 4
                                 
Example 7 :  3.5 x ?/0.7 = 1.75 
Solution. 1.75 x 0.7/3.5 = 1.225/3.5 = 0.35 

Example 8 : ? ÷ 25 ÷ 12 = 52.45 
Solution: ?/(25 x 12) = 5245
  => ? = 25 x 12 x 52.45 
  => ? = 15735  

Type 2

To Find the Approximate Value of (?)

Example 9  16.141 x 23.905 x ? = 432.2234
Solution: ? = 432.2234/(16.141 x 23.905) ≈ 432/(16 x 24) ≈ 1.125 ≈  1 

Example 10 : 936.118 ÷ 23.91 ≈  13.001 = ?
Solution : ? = 936.118/(23.91 x 13.001) ≈ 936/(24 x 13) ≈ 3 

Example 11 : 195.559 + 39.0937 x 41.679 - 24.591 = ?
Solution : ? ≈ 196 + (39 x 42) - 25 ≈ 196 + 1638 - 25 ≈ 1809