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How to Solve Terminating Fractions and Repeating Decimals Involving 9 in the Denominator

Updated on October 5, 2018
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Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

Changing the Fraction into a Decimal

Oftentimes in mathematics, we will have a fraction that when converted into a decimal will go on and on forever. Because of this never-ending nature to the number, we call them repeating decimals. An example is 0.1666666666… The fractional form of this, 1/6, can be thought of as terminating because the fraction does not require us to go on indefinitely and will be a simple ratio of two numbers, ending our search for the precise value and hence the use of the word terminates. We can just write the decimal as a ratio of two numbers and stop there.

When you are presented with a problem that will involve these conversions, it is nice to have a short-cut, and math is full of them. They are our friend so long as we understand why we are using it. And of course the easier the short-cut the better. Fortunately, when you have a fraction with a nine in the denominator, such an easy route exists.

Let us take a look at 1/9. Normally, when we convert a fraction into a decimal, we count how many times the denominator goes into the numerator. This is simply a division problem. But right away we know that 9 does not go into 1, so we must write that it goes into it 0 times. Now, we add a decimal behind the 0 and add a 0 to the 1, so now it is 10. 9 certainly goes into 10, and when you find the difference, it is 1 again. Applying this trick over and over again, we find that 1/9 = 0.1111111… or written in correct form 0.̅1. Now try 5/99. You will find that after division, the answer is 0.̅0̅5. So how do I go about this so that it is easier than division?

Simply follow this formula: 1. The numerator of the fraction will be the number that is in the decimal. 2. The number of 9’s in the denominator is the number of decimal places we will need. 3. Put a bar over the number that is behind the decimal place. In the case of 1/9, 1 is in the decimal and we have only one 9, so we move the decimal over by one. Now we have 0.1, and once we place a bar over it, becomes 0.̅1. For 5/99, 5 will be in the decimal, we have two 9’s so we move the decimal over two spots to make it 0.05, and finally throwing a bar over it makes it 0.̅0̅5.


Changing the Decimal into a Fraction

What if we have a repeating decimal and I want to rewrite it as a terminating fraction? We could simply work backwards from what we did above. For 0.̅1̅9, we have a 19 after the decimal point, so it will go in the numerator and sincewe have 2 decimal places, so that means we need two 9’s in the denominator, for a final answer of 19/99.

We can also use another trick to figure out the answer. Let’s make x the decimal we are trying to make into a fraction. If 0.̅7 is the decimal, then 0.̅7 = x. Multiply both sides by 10 gives us 7.̅7 = 10x. If I take away x from both sides, that gives me 7 = 9x, or x = 7/9. The easy trick to remember here is that the number of decimal places is the number of 10’s I need. Since there was only one in the above example, I only need to multiply both sides by one 10. For the one we did before, 0.̅1̅9, we need to multiply both sides by two 10’s, or 100, to solve the problem with this technique.

Now, try it out for yourself. Give the problems below a shot and see if you can get the right answer.

© 2013 Leonard Kelley


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