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How to Convert between Fractions and Repeating Decimals

To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number), using either calculator or pencil and paper. For example, here’s how to convert the fraction

image0.jpg

to a decimal:

image1.jpg

So,

image2.jpg

To convert a decimal to a fraction, begin by placing that decimal over the number 1. Then keep multiplying both the numerator (top number) and denominator (bottom number) by 10 until both are whole numbers. For example, here’s how to convert the decimal 0.13 to a fraction:

image3.jpg

In both of these cases, the decimal forms of these numbers are terminating decimals — that is, the decimal can be written exactly in a finite number of decimal places.

In other cases, however, a decimal is repeating — that is, it cannot be written exactly without the numbers repeating forever.

Converting fractions to repeating decimals

Every fraction can be written as a decimal, either terminating or repeating. To write a fraction as a decimal, divide the numerator by the denominator. For example, here’s how you convert the fraction

image4.jpg

to a decimal:

image5.jpg

You can see that this decimal will never end, but instead will repeat forever in a pattern of 1s and 8s. Therefore,

image6.jpg

The bar over the numbers 18 means that these numbers are repeated infinitely: 0.1818181818. . . .

As another example, convert the fraction

image7.jpg

to a decimal by dividing:

image8.jpg

As in the previous example, the pattern of numbers in the answer repeats itself and will do so indefinitely. So,

image9.jpg

Converting repeating decimals to fractions

Every repeating decimal can be written as a fraction. A quick trick for converting a repeating decimal is to place the repeating numbers in the numerator of a fraction over the same number of 9s, and then reduce if necessary. For example, here’s how you convert the repeating decimals

image10.jpg

image11.jpg

and

image12.jpg

to fractions:

image13.jpg

To gain insight into why this trick works, here is a step-by-step way to convert a repeating decimal to a fraction using algebra. Suppose you want to convert the decimal

image14.jpg

to a fraction. Begin by letting x equal

image15.jpg image16.jpg

This decimal has two repeating decimal places, so multiply both sides of this equation by 100 — that is, the number that brings the whole repeating pattern to the left side of the decimal point:

image17.jpg

Note that this decimal still repeats forever. Now, subtract the original equation from this one:

image18.jpg

This step may seem strange, because on the right side of the equation you’re subtracting an infinite decimal from an infinite decimal. But this process removes the repeating decimal from the equation. Now, to solve for x, just divide by 99:

image19.jpg

As you can see, the result shows that

image20.jpg
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