Question

How do you convert $1.65$ ( $65$ being repeated) to a fraction?

Answer 1

Hint: Problems on conversions of repeating decimals to fractions can be easily done by an algebraic method in just a few steps. We first assume the given number to be x. Then, we multiply it with 100 and after that we subtract x from it. Further simplifying the result of the last subtraction, we find the value of x which comes out to be the fractional form of the given number of repeating decimals.

Complete step by step solution:
The number we are given is
$1.\overline{65}$ (here $65$ is being repeated)
We can convert this number into its fractional form by the algebraic method in a few steps.
We assume the given number of repeating decimals to be x as
\[x=1.\overline{65}\]
As the number x has repeating decimals up to two digits, we multiply it with the power $2$ of $10$ i.e. $100$ as shown below
\[\Rightarrow 100x=165.\overline{65}\]
Now to eliminate the repeating decimals we must subtract x from the above expression i.e. \[100x\] as shown below
\[\Rightarrow 100x-x=165.\overline{65}-1.\overline{65}\]
$\Rightarrow 99x=165-1$
$\Rightarrow 99x=164$
Further dividing both sides of the above equation by \[99\] we get
$\Rightarrow x=\dfrac{164}{99}$
Therefore, we conclude that the given number $1.65$ ( $65$ being repeated) can be written in fraction form as $\dfrac{164}{99}$.

Note: While doing conversions of repeating decimals to fractions we must first see how many digits after the decimal points are being repeated from which we will decide the power of 10 to be multiplied with x. Also, the entire calculation of this type of problems must be done properly to avoid inaccuracy in the answer.