Which fractions stop, and what decides it

Divide 3 by 8: you get 0.375, which stops. Divide 5 by 12: you get 0.41666…, and the 6s go on. Which fractions stop?

Look at the primes in the bottom number

A fraction stops when its bottom number has only 2s and 5s. Cut the fraction down first, since a 3 may cancel. Then write the bottom number as a product of primes.

• 8 is 2 × 2 × 2. Only 2s. 3/8 stops.
• 40 is 2 × 2 × 2 × 5. Only 2s and 5s. 9/40 stops.
• 12 is 2 × 2 × 3. There is a 3. 5/12 does not stop.

A decimal that stops is a fraction over a power of ten. A power of ten is 10, 100, 1000 and so on. 0.375 is 375/1000.

Ten is 2 × 5. So a power of ten has only 2s and 5s.

Now take a fraction in lowest terms. Its bottom number must have just 2s and 5s. Only then can it equal a fraction over a power of ten. A 3 on the bottom cannot cancel. The top has no 3.

Example. In 17/125, the bottom number is 125 = 5 × 5 × 5. It has only 5s, so 17/125 stops. Multiply top and bottom by 8: 17/125 = 136/1000 = 0.136.

Try the lab to finish this lesson.