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 two-digit number be $10x + y$, where $x$ is the tens digit and $y$ is the units digit.
• The number formed by reversing the digits is $10y + x$.
• The given ratio is $\frac{10x + y}{10y + x} = \frac{4}{7}$.
• Cross-multiplying gives $7(10x + y) = 4(10y + x)$, which simplifies to $70x + 7y = 40y + 4x$.
• Rearranging terms yields $66x = 33y$, or $2x = y$.
• Since $x$ and $y$ are digits ($1 \le x \le 9$, $0 \le y \le 9$) and $y = 2x$, the possible values for $x$ are 1, 2, 3, and 4 (since if $x=5$, $y=10$ which is not a digit).
• The corresponding pairs $(x, y)$ are $(1, 2), (2, 4), (3, 6), (4, 8)$, resulting in the numbers 12, 24, 36, and 48.
• There are exactly 4 such pairs, so the correct option is (b).
Why the other options are wrong
- (a) 5
- Option (a) is incorrect because it overcounts the valid pairs by including cases where the unit digit exceeds 9.
- (c) 3
- Option (c) is incorrect because it undercounts the valid pairs by missing one of the four valid integer solutions for x.
- (d) 2
- Option (d) is incorrect because it significantly undercounts the valid pairs, ignoring multiple valid digit combinations.
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.