Question

How do you write the repeating decimal $0.175$ where 175 is repeated as a fraction?

Answer 1

Hint: Here, we are required to write the repeating decimal $0.175$ where 175 is repeated as a fraction. Thus, we will assume the given decimal number as $x$ and then, we will multiply both sides with a number such that the repeating digits get transferred to the left side of the decimal point. Subtracting them, we will be able to get a number without the repeating digits and hence, we will be able to find the simplest possible form of fraction of the given decimal number.

Complete step by step solution:
According to the question, we are given a decimal number such that the terms after the decimal point are being repeated.
Thus, this can be written as: $0.\overline{175}$
This bar represents that actually this decimal number is repeating these three digits again and again.
Thus, let us assume,
$x=0.\overline{175}$…………………………..$\left(1\right)$
Now, as we can observe, the decimal point is after 3 digits. Thus, we will put three zeroes in the denominator. Hence, we will actually divide this by 1000 to convert the decimal number to fractional number.
Hence, in order to obtain the same repeating fraction after the decimal point, let us multiply both sides by $\left(1\right)$ by 1000.
Thus, we get,
$1000x=175.\overline{175}$……………………$\left(2\right)$
Now, subtracting $\left(1\right)$ from $\left(2\right)$, we get,
$1000x-x=175.\overline{175}-0.\overline{175}$
$\Rightarrow 999x=175$
Dividing both sides by 999, we get,
$\Rightarrow x={\displaystyle \frac{175}{999}}$
Clearly, this cannot be simplified further.

Hence, the repeating decimal $0.175$ where 175 is repeated as a fraction can be written as ${\displaystyle \frac{175}{999}}$ in the simplest form.
Therefore, this is the required answer.

Note:
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. The infinitely repeated digit sequence is called the repetend or reptend. We represent the recurring digits with the help of either a bar or by applying some dots (…).