Question

How do you convert $0.789$ ( $789$ repeating) to a fraction?

Answer 1

Hint: The process for converting repeating decimals to fractions can be confusing at first to us, but with the algebraic method we can convert it into its fractional form easily. We first assume the given number to be x. After that we multiply it with $1000$ as we have $3$ digits repeating after the decimal point. Then, we subtract $x$ from $$1000x$$ and upon doing that we simplify the expression which we got from the subtraction. This will give us the fractional form of the given number.

Complete step by step solution:
The number we are given is
$0.\overline{789}$ (here the digits $789$ is being repeated)
We can convert this number into its fractional form by the algebraic method in few steps as shown below.
We assume the given number of repeating decimals to be x as
$$x=0.\overline{789}$$
As the number x has repeating decimals up to three digits, we multiply it with the power $3$ of $10$ i.e., $1000$ as shown below
$\Rightarrow 1000x=789.\overline{789}$
Now to eliminate the repeating decimals we must subtract x from the above expression i.e., $$1000x$$ as shown below
$\Rightarrow 1000x-x=789.\overline{789}-0.\overline{789}$
$\Rightarrow 999x=789.0-0.0$
$\Rightarrow 999x=789$
Further dividing both sides of the above equation by $$999$$ we get
$\Rightarrow x={\displaystyle \frac{789}{999}}$
Therefore, we conclude that the given number 0.789 (789 being repeated) can be written in fraction form as ${\displaystyle \frac{789}{999}}$ .

Note: We have to keep in mind that while doing conversions of repeating decimals to fractions we must first see how many digits after the decimal points are being repeated so that we can decide the power of $10$ to be multiplied with x which is $3$ in this case. Also, the rest of the calculation of this problem must be done properly so that inaccuracy in answer can be prevented.