**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.

**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

Hence, 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

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

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

(2) - (1)

aaaaaaaaaaaaaaaaa100 x = 47.4747.........aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaax = 0.4747...............aaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaa99 x = 47.0000aaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa

x = 47/99

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

**Example 4 :**

Convert the following repeating decimal as fraction

0.57777..........

**Solution :**

Let x = 0.5777777............... ------(1)

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

10 x = 0.57777777.............. ------(2)

(2) - (1)

aaaaaaaaaaaaaaaa10 x = 5.77777777.........aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaax = 0.57777777...............aaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaa9 x = 5.20000aaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa

x = 5.2/9

Multiply the numerator and denominator by 10

x = 52/90

x = 26/45

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

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