Question

How do you convert 0.148 (148 repeatings) to a fraction?

Answer 1

Hint: First of all, let us assume 0.148148148…. as $ x $ and mark it as eq. (1) then multiply 1000 on both the sides and mark this as eq. (2) then subtract eq. (1) from eq. (2) and then solve the subtraction to convert the recurring decimal to fraction.

Complete step by step answer:
The recurring decimal that we have to convert to a fraction is as follows:
0.148148148…….
Now, let us assume the above recurring decimal as x then the above recurring decimal will look like:
 $ x=0.148148148.......... $ ……… Eq. (1)
Multiplying 1000 on both the sides in the above equation we get,
 $ 1000x=148.148148........ $ ………… Eq. (2)
Now, we are going to subtract eq. (1) from eq. (2) and we get,
 $ \begin{align}
  & 1000x=148.148148..... \\
 & \dfrac{-x=-0.148148148.......}{999x=148.000} \\
\end{align} $
Rewriting the above equation we get,
$\Rightarrow$ $ 999x=148 $
Dividing 999 on both the sides we get,
$\Rightarrow$ $ \dfrac{999x}{999}=\dfrac{148}{999} $
On L.H.S of the above equation, 999 will be cancelled out from the numerator and the denominator and we get,
$\Rightarrow$ $ x=\dfrac{148}{999} $
In the above, as we have assumed x as the recurring decimal and in the above equation we have converted the decimal to a fraction and its value is $ \dfrac{148}{999} $ .

Note:
 The trick to converting a recurring decimal is to multiply the decimal with the power of 10 in such a way so that the recurring number in the decimal will be canceled out. For instance, in this problem, we have multiplied the given recurring decimal with 1000 so that point will be shifted to three numbers 148 and then when we subtract this multiplication with the original recurring decimal then we have found that the recurring number “148” has been canceled out.