The ratio of a two-digit natural number to a number formed by reversing its digits is 4 : 7. The number of such pairs is
- (a) 5
- (b) 4 ✓ UPSC's answer
- (c) 3
- (d) 2
Why the answer is (b)
• Let the number be 10x + y; the number with its digits reversed is 10y + x.
• (10x + y) : (10y + x) = 4 : 7 gives 70x + 7y = 40y + 4x, so 66x = 33y, that is y = 2x.
• Since y must be a single digit, x can be 1, 2, 3 or 4, giving 12, 24, 36 and 48 (x = 5 would make y = 10).
• Check: 12 : 21, 24 : 42, 36 : 63 and 48 : 84 are all 4 : 7. So there are 4 such pairs, option (b).
Why the other options are wrong
- (a) 5
- A fifth pair would need x = 5 and y = 10, which is not a digit.
- (c) 3
- 3 misses one of the four pairs 12, 24, 36 and 48.
- (d) 2
- 2 misses two of the four pairs 12, 24, 36 and 48.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.