Determine Whether the Given Rational Numbers Have Terminating or Non-Terminating Repeating Decimal Expansions
Video Explanation
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Solution
Question: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Important Rule
A rational number in its lowest form has a terminating decimal expansion if and only if the prime factorisation of its denominator has only the primes 2 and/or 5.
Otherwise, it has a non-terminating repeating decimal expansion.
(i) 23/8
8 = 23
The denominator has only the prime factor 2.
∴ 23/8 has a terminating decimal expansion.
(ii) 125/441
441 = 21 × 21 = 32 × 72
The denominator contains primes other than 2 and 5.
∴ 125/441 has a non-terminating repeating decimal expansion.
(iii) 35/50
35/50 = 7/10 (in lowest form)
10 = 2 × 5
The denominator has only the primes 2 and 5.
∴ 35/50 has a terminating decimal expansion.
(iv) 77/210
77/210 = 11/30 (in lowest form)
30 = 2 × 3 × 5
The denominator contains prime 3 in addition to 2 and 5.
∴ 77/210 has a non-terminating repeating decimal expansion.
(v) 129 / (22 × 57 × 717)
22 = 2 × 11
57 = 3 × 19
717 = 3 × 239
The denominator contains primes other than 2 and 5.
∴ The given rational number has a non-terminating repeating decimal expansion.
(vi) 987/10500
10500 = 22 × 3 × 53 × 7
The denominator contains primes other than 2 and 5.
∴ 987/10500 has a non-terminating repeating decimal expansion.
Final Answer
(i) Terminating
(ii) Non-terminating repeating
(iii) Terminating
(iv) Non-terminating repeating
(v) Non-terminating repeating
(vi) Non-terminating repeating
Conclusion
Thus, by examining the prime factorisation of the denominators in lowest form, we can decide the nature of the decimal expansion without performing long division.