Question

How do you convert 1.23 (23 being repeated) to a fraction?

Answer 1

Hint: For the given question we are given to convert 1.23 (23 being repeated) to a fraction. For that first of all we have to consider the given value as x and then we have to multiply the equation with 100. After that we have to subtract the starting equation from the last equation. So, therefore after simplifying the above process we can convert into fraction.


Complete step by step answer:
For the given question we have to convert \[1.23\](23 being repeated) to a fraction.
So, let us consider the given equation as equation (1).
\[x=1.2323232323........\text{ }.....\text{(1)}\]
Since x is recurring with 2323….. we have to multiply the equation (1) with 100 to the equation (1).
By multiplying equation (1) with 100 we get
\[\Rightarrow 100x=123.23....\]
Let us consider the above equation as equation (2).
\[100x=123.23....\text{ }........\left( 2 \right)\]
Now for the value of ‘x’ in fraction we have to subtract equation (2) from the equation (1).
By subtracting equation (2) from the equation (1), we get
\[\Rightarrow 100x-x=123.23.....\text{ -1}\text{.23}.........\]
By simplifying, the above equation we get
\[\Rightarrow \text{99x=122}\]
Let us consider the above equation as equation (3), we get
\[\Rightarrow \text{99x=122}.......\left( 3 \right)\]
By dividing with 99 on both, sides we get
\[\Rightarrow \text{x=}\dfrac{122}{99}\]
Let us consider the above equation as equation (4).
\[\Rightarrow \text{x=}\dfrac{122}{99}.........\left( 4 \right)\]
Therefore by converting the value of ‘x’ from recurring decimal to fraction we get-
\[\Rightarrow \text{x=}\dfrac{122}{99}\].

Note: The main logic of the problem is we have to multiply the equation (1) with 100 so therefore decimal will be eliminated and now we can get the fraction easily. If the question is given as 3 numbers recurring (ex: 1.235235……., 568.598598598…..) then we have to multiply it with 1000.