Question

Convert the given recurring decimals into fraction
$1.\overline{237}$

Answer 1

Hint: Now to solve this question consider $x=1.\overline{237}$ . To this equation multiply 1000 on both sides to get the second equation. Once we have two equations then subtract the two equations to get the third equation. Now divide the obtained equation by 999 to find the value of x.

Complete step-by-step answer:
Now the given recurring decimal is $1.\overline{237}$ . Now by recurring we mean repeating hence when we talk about recurring decimals we mean the decimals are repeating. Now note that the numbers below the bar sign will only repeat. For example if the decimal is $2.4\overline{67}$then this means the decimal is $2.4676767.....$
Now since our decimal is $1.\overline{237}$ we have 237 under the bar sign.
Hence 237 will repeat itself
Which means our decimal is $1.237237237.....$
Now also note that we have rational number as either recurring decimals or terminating decimals
Hence we have $1.\overline{237}$ is a rational number
And since $1.\overline{237}$ it can be written in a fractional form.
Now let us try to write it in fractional form
Let us consider $x=1.237237237...$
Multiplying the whole equation by 1000 on both sides we get
$1000x=1237.237237...$
Now subtracting equation $1000x=1237.237237...$ from equation $x=1.237237237...$ we get
$1000x-x=1237.237.237....-1.237237...$
$\Rightarrow 999x=1236$
Now dividing the equation by 999 we get
$x=\dfrac{1236}{999}$
Hence from equation (1) we have
$1.237237237...=\dfrac{1236}{999}$
Hence the given decimal is written in fractional form.

Note: Now we can write any recurring decimal with this method. All we have to do is assume x = given recurring decimal and then multiply the equation with ${{10}^{n}}$ where n is the number of terms after decimal once it is written in bar form.
For example if the decimal is $1.2\overline{7}$ we will multiply it by 100.