Question

How do you write $1.123$ repeating as a fraction?

Answer 1

Hint: The number given is called a number with a recurring decimal part. The recurring part is denoted by a bar on the top of the number that recur. The given decimal number can be written as $1.\overline{123}$. Consider the number to be x and multiply the number with 1000 and subtract with x and find the value of x.

Complete step by step solution:
In the question, the decimal number is given as 1.123 repeating. When we say that 1.123 is repeating, we mean that the number is a non-terminating decimal number and the digits after decimal point (that are 123) repeat in the same order. The number is called a number with a recurring decimal part. The recurring part is denoted by a bar on the top of the number that recur. The given decimal number can be written as $1.\overline{123}$.

Or we can also write it as $1.123123123.....$
Now, let us write this decimal number as a fraction. For that, let assume the number to be x.
i.e. $x=1.123123123.....$ ….(i)
Now, multiply equation (i) with 1000.
Then,
$1000x=1123.123123123.....$ ….. (ii)
Now, subtract (i) form (ii).
$1000x-x=1123.123123.....-1.123123.......$
$\Rightarrow 999x=1122$
$\therefore x=\dfrac{1122}{999}=\dfrac{374}{333}$.

Therefore, the fractional form of the decimal number $1.123$ repeating is equal to $\dfrac{374}{333}$.

Note: Note that we have to multiply the variable $x$ with a factor $10$ raised to a whole number. The whole number in the exponent place must be equal to the number of decimal digits in the recurring part. In this case, the number of digits that were recurring is 3.