Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 720, 415 and 13250. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.
Answer: 720 (20 = 22 × 5) terminates: 0.35. 415 (15 = 3 × 5) repeats: 0.26. 13250 (250 = 2 × 53) terminates: 0.052.
Step-by-step solution
Idea: Each fraction is already in lowest terms. Factorise the denominator: only 2s and 5s → terminating; any other prime → repeating. Then confirm by dividing.
- 20 = 22 × 5 and 250 = 2 × 53: only 2s and 5s, so 720 and 13250 terminate. 15 = 3 × 5 has the prime 3, so 415 repeats.1 mark
- 720: 70 ÷ 20 = 3 r 10; 100 ÷ 20 = 5 r 0 → 0.35 ✓ (terminates).½ mark
- 415: 40 ÷ 15 = 2 r 10; 100 ÷ 15 = 6 r 10; 100 ÷ 15 = 6 r 10 … → 0.2666… = 0.26 ✓ (repeats).1 mark
- 13250: 130 ÷ 250 = 0 r 130; 1300 ÷ 250 = 5 r 50; 500 ÷ 250 = 2 r 0 → 0.052 ✓ (terminates).½ mark
7/20 = 0.35 (terminating), 4/15 = 0.26 (repeating), 13/250 = 0.052 (terminating).
Check: 0.35 × 20 = 7 ✓; 0.052 × 250 = 13 ✓; 0.2666… × 15 = 3.999… = 4 ✓.
Answer to write in the exam
20 = 22 × 5 ⇒ 720 terminating
15 = 3 × 5 ⇒ 415 non-terminating repeating
250 = 2 × 53 ⇒ 13250 terminating
Long division: 720 = 0.35
415 = 0.2666… = 0.26
13250 = 0.052
∴ 720 and 13250 terminate; 415 repeats.
Common mistakes that cost marks
- Saying 415 terminates because 15 has the factor 5. Every prime factor must be 2 or 5; the 3 makes it repeat.
- Writing 13250 = 0.52, losing the 0 in the first decimal place.
- Writing 415 = 0.26: only the 6 repeats.
How this can come in the exam
MCQ (1 mark)
Which of these has a non-terminating repeating decimal expansion?
- 1780
- 9125
- 1124
- 316
Show answer
(C) 1124
24 = 23 × 3 contains 3; 80, 125 and 16 contain only 2s and 5s.
Short answer (2 marks)
Without long division, say whether 2156 terminates. Then write its decimal.
Show answer
2156 = 38 in lowest terms and 8 = 23, so it terminates (1 mark). 38 = 0.375 (1 mark).Try one yourself
Decide without dividing, then check: 940 and 518.
Show answer
40 = 23 × 5: terminates, 0.225. 18 = 2 × 32: repeats, 0.27.
More questions like this
- Perform the long division for 113. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 213? Now compute 313, 413, etc. What do you notice?
- Classify the following numbers as rational or irrational:
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