Activity: Repeating and Terminating Decimals

Every fraction (i.e. rational number) can be written in decimal form.

So how do you rename fractions (the terminating kind) as decimals?

First, find an equivalent fraction that has 10, 100, 1000, etc. in its denominator. For example.

1/2 = 5/10

3/4 = 75/100

5/8 = 625/1000

Quick Review “Cloning” fractions
Two fractions are clones if they are equivalent.
1/2 is the clone of 5/10. Why? If you multiply numerator and denominator by the same number you have a cloned or equivalent fraction.

Another way to write fractions that have 10, 100, 1000, etc in the denominator is to write it in decimal notation.

75/100 is read “seventy five hundredths” and equals .75

BTW - If you say .75 with meaning you say “75 hundredths”, not “point 75”

Repeating decimals
Some fractions are not as nice. They have numbers that repeat in their decimal expression. For example,

1/3 can’t be cloned to have 10, 100. 1000, etc. in its denominator. Why? Because you can’t come up with a whole number to clone with for 3 since 3 does not divide into 10, 100, 1000, etc. evenly.

So the best we can do is to treat 1/3 as a division problem (which it is) and do the division.

0.3333333
3 | 1.0000000

1/3 = 0.3333333......

Here 3 is the repeating number.

Some fractions repeat more than 1 number. For example, try

1/11

You should see that 0 and 9 repeat.

Activity: Use a spreadsheet to change some common fractions to decimals and determine if they are repeating or terminal and how many numbers are repeating.

Create the spreadsheet below:

Click on Spreadsheet above to download it.

Handy Java tool for determining whether a fraction terminates or not.

For more on this tool see repeating or terminating? by Jeff LeMieux