A student asks about repeating decimals

A student asks,

How do I convert repeating decimal like 21.6666 repeater into a fraction?

It helps to know which fractions tend to give you repeating decimals. Anything in the thirds family may give you a repeater. Remembering that the ‘/’ symbol also means divide, put


into your calculator and you’ll get

0.3333 repeater

Do the same for 1/6 and you’ll get 0.16666 repeater, which you may have noticed is one-half of 0.33333 repeater.

So 0.66666 repeater is twice as big as 0.33333 repeater. Therefore 0.66666 repeater is twice one third. That gives us two thirds or 2/3. Put that into your calculator to verify.

Now we know that 21.66666 repeater is equal to

21 2/3

It is also helpful to note that the sevenths family of numbers also tend to be repeaters. Put 1/7 into your calculator and you will get


Note that in the case you’d need to draw a line over the 142857 as that’s the part that repeats.