Predict before you divide

The idea

For p/q in lowest terms, the decimal expansion terminates precisely when the prime factors of q are only 2, only 5, or both 2 and 5, since the denominator can then be made a power of 10.

Can we predict?

3/20 = 0.15 stops. 1/3 = 0.333… never stops. Can you tell which fractions stop without doing the long division?

Will the decimal of 7/8 stop or repeat?

• It stops
• It repeats
• I cannot tell without dividing

Stop or repeat?

Look at the prime factors of q. Green means 2 or 5, red means any other prime. Choose, then the toy checks.

2s and 5s

Let p/q be a rational number in lowest terms. Its decimal terminates precisely when the prime factors of q are only 2, only 5, or both 2 and 5, because then the denominator can be made a power of 10.

For 3/20: 20 = 2² × 5. Multiply the top and the bottom by 5 and you get 15/100, which is 0.15.

Notes

For p/q in lowest terms, the decimal terminates precisely when the prime factors of q are only 2, only 5, or both 2 and 5, since the denominator can then be made a power of 10.

Check yourself

Which fraction, in lowest terms, has a terminating decimal?

7/25: multiply the top and the bottom by 4 to make the bottom 100. What is the new numerator?

Answer: 28

7 × 4 = 28 and 25 × 4 = 100, so 7/25 = 28/100 = 0.28.

Which of these fractions, in lowest terms, has a repeating decimal?

3/12 is not in lowest terms. Does its decimal stop?

• 7/16 — correct. 16 = 2 × 2 × 2 × 2. Only 2s, so it can be made a power of 10: 7/16 = 0.4375.
• 4/9. 9 = 3 × 3. The factor 3 can never become a power of 10.
• 5/6. 6 = 2 × 3. The factor 3 can never become a power of 10.
• 9/40. 40 = 2 × 2 × 2 × 5. Only 2s and 5s, so 9/40 = 0.225 stops.
• 3/8. 8 = 2 × 2 × 2. Only 2s, so 3/8 = 0.375 stops.
• 5/12 — correct. 12 = 2 × 2 × 3. The factor 3 means the decimal repeats: 5/12 = 0.41666…
• No, because 12 has the prime factor 3. Reduce first: 3/12 = 1/4. The factor 3 cancels.
• Yes. 3/12 = 1/4 = 0.25 — correct. Yes. In lowest terms the denominator is 4 = 2 × 2, so it stops.
• We cannot tell. We can: first write the fraction in lowest terms.