Question

How do you find the repeating decimal $0.427$ with $427$ repeated as a fraction?

Answer 1

Hint: Consider the given number to be some constant (say x) and then multiply both sides by raising $10$ to the power of the number of digits repeated after decimal point (if two digits repeat then raise the power of $10\;{\text{to}}\;2$ ), then subtract the second equation from the original equation and then divide both sides with x coefficient, you will get the desired result.

Complete step by step solution:
We have to take several steps to convert $0.427$ with $427$ repeated to a fraction. Repeating or recurring decimals have their own way of being transformed into a fraction.
We have to assume the value of $0.427$ with $427$ repeated in the first step to be $x$
$
   \Rightarrow x = 0.427\;(427\;{\text{being}}\;{\text{repeated}}) \\
   \Rightarrow x = 0.427427427...\; - - - - - - (i) \\
 $
Now we can see in the above equation that three digits i.e. $4,\;2\;{\text{and}}\;7$ are being repeated, so we will multiply the equation $10$ raise to the power of $3$ (Number of digits being repeated after decimal point).
So multiplying by ${10^3} = 1000$ to both the sides,
$
   \Rightarrow 1000 \times x = 1000 \times 0.427427427... \\
   \Rightarrow 1000x = 427.427427427...\; - - - - - - (ii) \\
 $
Now subtracting equation (i) from equation (ii), we will get
$
   \Rightarrow 1000x - x = 427.427427427... - 0.427427427... \\
   \Rightarrow 999x = 427.0000000... \\
 $
Since only $0$ is repeating in the decimal, so we can remove $0$ and write $427.0000000... = 427$
$ \Rightarrow 999x = 427$
Dividing both sides with coefficient of $x = 999$ to get the value of $x$
$
   \Rightarrow \dfrac{{999}}{{999}}x = \dfrac{{427}}{{999}} \\
   \Rightarrow x = \dfrac{{427}}{{999}} \\
 $

Therefore the required fraction of repeating numbers is $0.427427427... = \dfrac{{427}}{{999}}$

Note: When more than one digit will repeat after the decimal part then in order to convert that decimal into fraction number, when multiplying with its power of $10$, raise its power by the number of digits that are being repeated after the decimal point.