Q1.
Can you tell for which rational numbers the decimal will be terminating?
Answer
Exactly those p/q which, in lowest terms, have a denominator whose only prime factors are 2 and 5.
q = 2^m × 5^n → decimal terminates
q has any other prime factor → decimal repeats
q has any other prime factor → decimal repeats
| Rational number | Denominator factorised | Decimal |
|---|---|---|
| 3/8 | 2³ | 0.375, terminates |
| 3/20 | 2² × 5 | 0.15, terminates |
| 13/250 | 2 × 5³ | 0.052, terminates |
| 4/15 | 3 × 5 | 0.2666…, repeats |
| 5/11 | 11 | 0.4545…, repeats |
Why it happens: a decimal that stops after k places is a fraction with denominator 10^k = 2^k × 5^k. So p/q terminates only if p/q can be rewritten with a denominator that is a power of 10 — and multiplying q by anything can never remove a prime factor of 3, 7, 11, … already sitting inside it. Conversely, if q = 2^m 5^n, multiplying top and bottom by the missing power of 2 or 5 turns the denominator into 10^max(m, n).
Tip: reduce to lowest terms first. 6/15 looks as if it has the prime 3, but 6/15 = 2/5 = 0.4, which terminates.