D is a 3-digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the digit at the hundred's place and the digit at the unit's place of D ?
- (a) 0
- (b) 7
- (c) 8 ✓ UPSC's answer
- (d) 9
Why the answer is (c)
• For D = 100a + 10b + c, the ratio D ÷ (a + b + c) is smallest when the hundreds digit is small and the other two digits are large.
• With hundreds digit 1, the number 199 gives 199 ÷ 19 ≈ 10.47; neighbours give more — 189 ÷ 18 = 10.5, 198 ÷ 18 = 11, 188 ÷ 17 ≈ 11.06.
• A hundreds digit of 2 or more pushes the ratio above 14 (e.g. 299 ÷ 20 ≈ 14.95), so D = 199 gives the least ratio.
• The hundreds digit is 1 and the units digit is 9, a difference of 8 — option (c).
Why the other options are wrong
- (a) 0
- The difference is 8, not 0, as the digits are 1 and 9.
- (b) 7
- The difference is 8, not 7, as the digits are 1 and 9.
- (d) 9
- The difference is 8, not 9, as the digits are 1 and 9.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.