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?

To find the 3-digit number D such that the ratio of the number to the sum of its digits is least, let the number be D = 100a + 10b + c, where a, b, c are its digits. The sum of its digits is S = a + b + c. We want to minimize the ratio R = D/S = (100a + 10b + c) / (a + b + c). 1. **Analyze the ratio:** The ratio R can be seen as a weighted average. The digit 'a' (hundreds place) contributes 100 times its value to the number D, while contributing only 1 time its value to the sum S. Similarly, 'b' (tens place) contributes 10 times to D and 1 time to S, and 'c' (units place) contributes 1 time to D and 1 time to S. To minimize the overall ratio D/S, we should: * Minimize the digit with the highest "weight" in the numerator (i.e., 'a', which is multiplied by 100). * Maximize the digits with lower "weights" in the numerator (i.e., 'b' and 'c', which are multiplied by 10 and 1 respectively), as they contribute more to the denominator (sum of digits) relative to the numerator (number value). 2. **Determine the digits:** * **Hundreds digit (a):** Since D is a 3-digit number, 'a' cannot be 0. To minimize the ratio, 'a' should be as small as possible, so a = 1. * **Tens digit (b):** To maximize the sum of digits and minimize the ratio, 'b' should be as large as possible, so b = 9. * **Units digit (c):** Similarly, 'c' should be as large as possible, so c = 9. 3. **Form the number D:** With a=1, b=9, and c=9, the number D is 199. 4. **Calculate the ratio for D=199:** Sum of digits S = 1 + 9 + 9 = 19. Ratio R = 199 / 19 = 10.473... (To confirm this is indeed the least, consider other numbers: If D=198 (a=1, b=9, c=8), S=18, R=198/18 = 11. If D=189 (a=1, b=8, c=9), S=18, R=189/18 = 10.5. If D=299 (a=2, b=9, c=9), S=20, R=299/20 = 14.95. The number 199 indeed gives the least ratio.) 5. **Calculate the required difference:** The digit at the hundred's place is 'a' = 1. The digit at the unit's place is 'c' = 9. The difference between these digits is |1 - 9| = |-8| = 8. The final answer is C) 8.