It terminates: The division eventually leaves a remainder of 0. The decimal stops. 38 = 0.375. (Can you tell for which rational numbers the decimal will be terminating?)
Answer: 3 ÷ 8: remainders 6, 4, 0, so 38 = 0.375 and the division stops. The decimal terminates exactly when the denominator (in lowest terms) has no prime factors other than 2 and 5; here 8 = 23.
Step-by-step solution
Idea: Long division of 3 by 8 ends with remainder 0. That happens because 8 = 23 divides 1000: 38 = 3751000. Any denominator made only of 2s and 5s can be built up to a power of 10 in the same way.
- Long division: 30 ÷ 8 = 3 remainder 6; 60 ÷ 8 = 7 remainder 4; 40 ÷ 8 = 5 remainder 0. The remainder is 0, so the division stops: 38 = 0.375.1 mark
- Why it stops: 8 = 23, and 38 = 3 × 5323 × 53 = 3751000. So the decimal terminates for those rational numbers pq (in lowest terms) whose denominator q has only 2 and/or 5 as prime factors.1 mark
3/8 = 0.375. A rational number in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5.
Check: 0.375 × 8 = 3 ✓.
Answer to write in the exam
30 ÷ 8 = 3 r 6; 60 ÷ 8 = 7 r 4; 40 ÷ 8 = 5 r 0
∴ 38 = 0.375 (terminating)
38 = 3 × 5323 × 53 = 3751000
∴ Terminating when the denominator (lowest terms) has no prime factors other than 2 and 5.
Common mistakes that cost marks
- Judging by the numerator. Only the denominator (in lowest terms) decides: 36 = 12 terminates even though 6 has the factor 3.
- Writing 38 = 0.38 after rounding. The exact decimal is 0.375.
- Thinking ‘even denominators terminate’. 16 = 0.1666… does not, because of the factor 3.
How this can come in the exam
MCQ (1 mark)
Which of these has a terminating decimal expansion?
- 56
- 715
- 940
- 221
Show answer
(C) 940
40 = 23 × 5; the others have the prime factor 3 (or 7) in the denominator.
Short answer (2 marks)
Write 1116 as a decimal by making the denominator a power of 10.
Show answer
16 = 24, so multiply top and bottom by 54 = 625: 687510000 (1 mark) = 0.6875 (1 mark).Try one yourself
Find the decimal expansion of 78 by long division.
Show answer
70 ÷ 8 = 8 r 6; 60 ÷ 8 = 7 r 4; 40 ÷ 8 = 5 r 0, so 78 = 0.875.
More questions like this
- It repeats: The division never reaches a remainder of 0, but the sequence of digits begins to loop infinitely. 511 = 0.454545 … = 0.45.
- Try to find the decimal expansions of 103 and 1112. What do you observe about the repetition of the digits after the decimal point?
- Why do some rational numbers have repeating decimal representations? Imagine calculating 17 using long division. You are dividing by 7. What are the possible remainders at each step? They can only be 1, 2, 3, 4, 5, or 6 (why not 0?).
- The decimal expansion of pq will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why?
- Convert 0.35 into the form pq.