Decimals and Fractions: Theory & Concepts

Decimal fraction: A fraction in which denominator is 10 or it is 102,103 etc. i.e. power of 10 is called decimal fraction e.g. 3/10, 3/100 they can be represented as 0.3, 0.03 etc.
There are two types of Decimals
  1. Non recurring decimal or terminating decimal
  2. Recurring decimal or non terminating decimal
1. Non-recurring or terminating Decimal: As the name suggests, non-recurring means which do not reoccur or repeat i.e. they terminate or come to end e.g. 0.5, 0.25, and 0.125 etc. In order to convert these fractions, just the numerator is divided by the denominator. For example if you need to convert 1/4 into decimal, divide 1 by 4 and get the answer as 0.25.
2. Recurring or non-terminating Decimal: As the name suggests recurring means which reoccur or repeat i.e. they do not terminate or come to end e.g. 0.333…, 0.545454…. etc. In order to convert these fractions, just the numerator is divided by the denominator. For example if you need to convert 1/3 into decimal, divide 1 by 3 and get the answer as 0.333...
These are further of two types
  1. Pure
  2. Mixed
Let us discuss each of them one by one:
  1. Pure recurring decimals: This is the type of recurring decimal in which all digits or set of digits (after decimal point) repeat .e.g. 0.7777…0r 2.373737….. . It can be simply written as or
  2. Mixed recurring decimals: This is the type of recurring decimal in which one or more digits after the decimal do not repeat while remaining digits or set of digits (after decimal point ) repeat .e.g. 1.4232323…
Let us learn now how to convert decimal into fractions i.e. recurring decimal into vulgar fraction. This will be better understood with the help of examples.
Example 1: Convert the given decimal to fraction
Sol: Let x = 0.777777 …..(i)
Multiplying both sides by 10 we get
10x = 7.7777….(ii)
Subtract equation (i) from (ii) we get
9x = 7 ⇒ x = 9/7.
Shortcut: Check the number of digits repeated and just write the same number of nines in the denominator e.g. in only one digit i.e. only the digit 7 is repeated so answer is 7/9 and in two digits i.e. 7 and 3 is repeated so answer is 73/99 and so on.

To convert mixed recurring decimal fraction into vulgar fraction

Example 2: Convert into vulgar fraction.
Sol: Let x = 0.2171717…. …..(i)
Multiplying both sides by 10 we get
10x = 2.171717……(ii )
Multiplying both sides of equation (ii) by 100 we get
1000x = 217.171717……(iii )
Subtract equation (ii) from (iii) we get
990x = 215 ⇒ x = 215/990.

Shortcut:

  1. In numerator: Write the complete number after decimal digits and subtract from it the digits which are repeated. i.e. in subtract 2 from 217 to get 215 in numerator.
  2. In denominator: Check the number of digits repeated and not repeated then write the number of nines in the denominator equal to number of digits repeated followed by number of zeros equal to number of digits not repeated i.e. in , two digits 17 are repeated so two nines are there followed by one zero as only one digit 2 that is not repeated. So we get 990 in denominator. So the answer of is 215/990
Now one more concept you should understand that we take whole thing as 1 i.e. 100% and if we take apart of it or spent a fraction of it then remaining is obtained by subtracting it from 1 or in percentage from 100.
Example 3: After spending 1/7th of his income a man is left with Rs. 600. Find his total income.
Sol: Let total income = x
Money spent =1/7 x ⇒ Money left = x -1/7 x = 6/7x
So according to the question
6/7x = 600 ⇒ x = 700.
Shortcut: If we spent 1/7 of something, we are left with (1- 1/7) i.e. 6/7 part of it. So directly 6/7 of x = 600 ⇒ x = 700.