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?
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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.
| Fraction | Primes in the bottom | Only 2s and 5s? | The division |
|---|---|---|---|
| 3/8 | 2 × 2 × 2 | yes | 0.375, stops |
| 9/40 | 2 × 2 × 2 × 5 | yes | 0.225, stops |
| 5/12 | 2 × 2 × 3 | no, a 3 | 0.41666… |
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.