EXPRESSING DECIMALS AS RATIONAL NUMBERS

A rational number is any number that can be written as a ratio in the form a/b, where a and b are integers and b is not 0.

Examples of rational numbers are 2/3 and 1/5.

We all know that 6 is an integer. But 6 also can be considered as rational number. 

Because, 6 can be written as 6/1. 

We can express terminating and repeating decimals as rational numbers.

Let us look at some examples to understand how to express decimals as rational numbers.  

Example 1 :

Write the decimal 0.825 as a fraction in simplest form. 

Solution : 

The decimal 0.825 means “825 thousandths.” Write this as a fraction.

To write “825 thousandths”, put 825 over 1000.

825 / 1000

Then simplify the fraction.

Both 825 and 1000 are the multiples of 25. So, divide both the numerator and the denominator by 25.

(825 ÷ 25) / (1000 ÷ 25)  =  33 / 40

0.825  =  33 / 40

So, the fraction equal to 0.825 is 33/40.

Example 2 :

Write the decimal 0.12 as a fraction in simplest form. 

Solution : 

The decimal 0.12 means “12 hundredths.” Write this as a fraction.

To write “12 hundredths”, put 12 over 100.

12/100

Then simplify the fraction.

Both 12 and 100 are the multiples of 4. So, divide both the numerator and the denominator by 4.

(12 ÷ 4) / (100 ÷ 4)  =  3 / 25

0.12  =  3 / 25

So, the fraction equal to 0.12 is 3/25.

Example 3 :

Convert the following repeating decimal as fraction  

0.474747...........

Solution :

Let x  =  0.474747...............  ------(1)

Here, number of repeating digits = 2. So we have to multiply 100 on both sides.

100x  =  47.4747..............  ------(2)

(2) -  (1) :  

100x - x  =  (47.4747.........) - (0.4747..........)

99x  =  47

x  =  47/99

So, the fraction equal to 0.474747........... is 47/99.

Example 4 :

Convert the following repeating decimal as fraction  

0.57777..........

Solution :

Let x  =  0.57777........------(1)

Here, number of repeating digits = 1. So we have to multiply 10 on both sides.

10x  =  5.7777..........------(2)

(2) -  (1) : 

10x - x  =  (5.7777.........) - (0.57777..........)

9x  =  5.2

Divide each side by 9.

x  =  5.2/9

Multiply the numerator and denominator by 10.

x  =  52/90

x  =  26/45

So, the fraction equal to 0.57777........... is 26/45.

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