Converting Recurring Fractions

I. Introduction

If you were to convert a non-recurring decimal value to fraction, we know how to do that.

For example to convert $0.175$ to fraction we take the following steps:

$0.175 = \dfrac{0.175}{1}$

Then we multiply the numerator and denominator with 1 followed by as many zeros as required to eliminate the decimal point.

$= \dfrac{175}{1000}$

then we reduce the result to get our final result:

$= \dfrac{7}{40}$.

But we cannot use this for recurring fractions, because recurring fractions are never ending, so we do not know how many zeros are needed to eliminate the decimal point.

Let us try a different approach for this. Let us try to convert the recurring decimal $0.\overline{7}$ to fraction.

We know that $0.\overline{7}$ can be written as:

$0.77777\ldots$

Since we do not know how much our fraction will be let us call is $x$. So,

$x = 0.777777\ldots$

$\texttip{\Rightarrow}{follows that} x = 0 + 0.777777\ldots$ Let us call this equation 1.

We multiply both sides by $10$.

$10x = 7.777777\ldots$

$\texttip{\Rightarrow}{follows that} 10x = 7 + 0.777777\ldots$ Let us call this equation 2.

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If we subtract equation 1 from equation 2, we get:

$10x - x = 7 + 0.777777\ldots - 0.777777\ldots$

we see that the fraction portion on the right side cancels out, since we have $0.777777\ldots - 0.777777\ldots$

$\texttip{\Rightarrow}{follows that} 9x = 7$

$\texttip{\Rightarrow}{follows that} x = \dfrac{7}{9}$

So, we have:

$0.\overline{7} = \dfrac{7}{9}$

Let us try with another example. We will convert $0.\overline{27}$ to fraction.

We can write $0.\overline{27}$ as $0.272727\ldots$

Like before, we assign the value $x$ to this fraction, so:

$x = 0.272727\ldots$ Let us call this equation 1.

Multiplying with $10$, will not help here, since the fraction after decimal will become $0.7272\ldots$. We need the fraction part to be same as before so that they can cancel out. In this case we will need to multiply by $100$.

$\texttip{\Rightarrow}{follows that} 100x = 27.272727\ldots$ Let us call this equation 2.

$\texttip{\Rightarrow}{follows that} 100x = 27 + 0.272727\ldots$

Again we subtract equation 2 from equation 1 and cancel out the fraction part:

$100x - x = 27 + 0.272727\ldots - 0.272727\ldots$

$\texttip{\Rightarrow}{follows that} 99x = 27$

$\texttip{\Rightarrow}{follows that} x = \dfrac{27}{99}$

$\texttip{\Rightarrow}{follows that} x = \dfrac{3}{11}$

$\texttip{\Rightarrow}{follows that} 0.\overline{27} = \dfrac{3}{11}$

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Let us take one more example, a little more complex this time. We will convert $0.12\overline{37}$ to fraction. Notice that the recurring symbol is only on top of the $37$ part and not the whole fraction.

Let us begin by assigning $x = 0.12\overline{37}$

We should try to get two equations with the same value of the fraction part, for them to cancel out.

So first we multiply both side by 100 and we get:

$100x = 12.37373737\ldots$ This is our equation 1.

$\texttip{\Rightarrow}{follows that} 100x = 12 + 0.373737\ldots$

Then again we multiply both sides of the original equation by $10000$ and we get:

$10000x = 1237.373737\ldots$ This is our equation 2.

$\texttip{\Rightarrow}{follows that} 10000x = 1237 + 0.373737\ldots$

Now we subtract equation 1 from equation 2, and we get:

$10000x - 100x = 1237 - 0.373737\ldots - 12 - 0.373737\ldots$

$\texttip{\Rightarrow}{follows that} 9900x = 1237 - 12$

$\texttip{\Rightarrow}{follows that} 9900x = 1225$

$\texttip{\Rightarrow}{follows that} x = \dfrac{1225}{9900}$

$\texttip{\Rightarrow}{follows that} x = \dfrac{49}{396}$

So:

$0.12\overline{37} = \dfrac{49}{396}$

You can try a few examples of this by creating any recurring pattern, converting and verifying the result using a calculator.