Pete's Math 110: March 2011

Thursday, March 31, 2011

How to write repeating decimals as single fractions.

HOW TO WRITE REPEATING DECIMALS AS A SINGLE FRACTION

By Peter Barrow

Let’s write 1.8989898989… (a repeating decimal) as a single fraction. Before we do that, let’s observe a few things about fractions and repeating decimals. You’ll want a calculator.

Enter 1/9 on your calculator. You get 0.99999999…

Now enter 3/9. You get 0.33333333….

What do you think you’ll get when you type in 7/9? You’re right: 0.7777777777…

Now try 83/99. You should get 0.838383838383…

Hopefully you’re starting to see a pattern. If you were to enter 237/999 you’d get 0.237237237237…

So, back to 1.89898989… Let’s think of it not as 1.8989898989… but rather as 1+0.8989898989… because from our experiment above, we know we can write 0.89898989… as 89/99. So 1+0.8989898989…=1+(89/99). Now we’re just adding fractions, which we can do in our sleep. We need like denominators:

1+(89/99)=(99/99)+(89/99)=(99+89)/99=188/99. Type it in on your calculator if you don’t believe me.

This provides a much simpler way of converting repeating decimals into fractions.

Let’s try some other examples:

10.1234123412341234…=10+(1234/9999)=(99990/9999)+(1234/9999)=(1234+99990)/(9999)=(101224/9999)

5.77777777777… = 5+(7/9)=(45/9)+(7/9)=(45+7)/9=52/9.

3.34343434343434… = 3+(34/99)=(297/99)+(34/99)=(331/99)

Now you try some on your own. I won’t include the answer, because to check your answer you can just type in the fraction you get on a calculator and see if you’re right.

7.55555555…..

8.656565656565….

1.369369369369….