April 2013 Brain Teaser Solution

Q:Which unit fractions 1/2, 1/3, 1/4 etc have terminating decimals and which repeat? For example, 1/2 = 0.5 terminates but 1/3 = .3333etc and repeats.

How many unit fractions up to 1/1000 have terminating decimals?  What is the key to figuring this out?

A:  28
The first handful of fractions that terminate starting with 1/2 are:

1/2 = .5, 1/4 = .25, 1/8 = .125, 1/10 = .1

Notice the denominators are all composed of 2s and 5s

For example 4 = 2 x 2 and 8 = 2 x 2 x 2 and 10 = 2 x 5

This works becuase our number system is base 10 and base 10 is easy to express fractions as decimals if the factors of the denominator are 2s and/or 5s.

Here is a list of numbers that have 2 and/or 5 as their only factors:

2 4 5 8 10 16 

20 25 32 40 50 

64 80 100 125 128

160 200 250 256 320

400 500 512 625 640
800 1000