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?
Answer & explanation
Answer: (c) 8
The ratio (number ÷ digit sum) is smallest when the hundreds digit is as small as possible and the tens and units digits are as large as possible. 199 gives 199/19 ≈ 10.47, lower than 189/18 = 10.5 or any other, so D = 199 and the difference between 9 and 1 is 8.
- Write D = 100a + 10b + c. Raising a adds 100 to the number but only 1 to the digit sum, so a should be 1; b and c should be as large as possible.
- Take a = 1, b = 9, c = 9: 199/19 ≈ 10.47.
- Compare the nearest rivals: 189/18 = 10.5, 179/17 ≈ 10.53, 198/18 = 11 — all larger.
- So D = 199: hundreds digit 1, units digit 9, difference 9 − 1 = 8.
Remember · To minimise a number divided by its digit sum, keep the high-place digit smallest and push the digit sum into lower places; test close rivals.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·