Question
Download Solution PDFWhich of the following have non-terminating decimal expansions?
(a) \(\frac{987}{3150}\)
(b) \(\frac{54}{1125}\)
(c) \(\frac{133}{1680}\)
(d) \(\frac{19}{3125}\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula used:
A fraction has a terminating decimal expansion if its denominator, when expressed in its simplest form, has only 2 and/or 5 as prime factors.
Calculation:
(a) 987/3150 = 329/1050 = 329/(2 x 3 x 52 x 7).
Since the denominator has prime factors other than 2 and 5 (specifically 3 and 7), the decimal expansion is non-terminating.
(b) 54/1125 = 18/375 = 6/125 = 6/53.
Since the denominator has only 5 as a prime factor, the decimal expansion is terminating.
(c) 133/1680 = 19/240 = 19/(24 x 3 x 5).
Since the denominator has a prime factor other than 2 and 5 (specifically 3), the decimal expansion is non-terminating.
(d) 19/3125 = 19/55.
Since the denominator has only 5 as a prime factor, the decimal expansion is terminating.
∴ (a) 987/3150 and (c) 133/1680 have non-terminating decimal expansions.
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