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The simplest fraction for $0.1 63 ¯$ is ____.

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Detailed Solution

The simplest fraction for

0.1 63 ¯

955

.
To express it in the form of

p q

, assume this number as, say x, and write in the expanded form as follows,

x = 0 ⋅ 1636363.......

………... (i)
First shift 1 towards the left of the decimal, by multiplying each side of equation (1) by 10 as follows,

10 x = 1 ⋅ 636363.....

…………. (ii)
To shift the decimal point by two more places to the right, multiply the equation (ii) by 100 on each side as follows,

1000 x = 163 ⋅ 636363.....

…………(iii)
Now subtract equation (ii) from (iii) and simplify as follows,

1000 x − 10 x = 163.636363..... − 1.636363..... 990 x = 162 x = 162990 x = 955

From the above result, we can conclude that,

0.1 63 ¯ = 955

.
So, it is the simplest fractional form of the given repeating decimal.

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