A three-digit number has digits whose sum is 10. The middle digit is equal to the sum of the other two digits, and the number increases by 99 when its digits are reversed. What is this three-digit number?

Introduction / Context:
This question combines digit sums and digit reversal concepts in a three-digit number. Problems of this type are common in number system sections to test reasoning about place values, digit relationships and simple algebra. We are given conditions on the sum of digits, a relationship between the digits, and the effect of reversing the digits on the value of the number. Using these clues, we can form equations to determine the exact number.

Given Data / Assumptions:

• The number is a three-digit number with digits a, b and c.
• The sum of the digits is 10, so a + b + c = 10.
• The middle digit b equals the sum of the other two digits, so b = a + c.
• Reversing the digits increases the number by 99.
• We must find the original three-digit number.

Concept / Approach:
Let the three-digit number be written as 100a + 10b + c. Reversing the digits gives 100c + 10b + a. We are told that the reversed number is 99 more than the original number. This gives an equation linking a and c. Along with the condition on the digit sum and the relation b = a + c, we can solve for a, b and c. Finally, we match the resulting number against the options.

Step-by-Step Solution:
Step 1: Let the original number be 100a + 10b + c. Step 2: The sum of the digits is a + b + c = 10. Step 3: The middle digit equals the sum of the others, so b = a + c. Step 4: Substitute b into the sum: a + (a + c) + c = 10, so 2a + 2c = 10, giving a + c = 5. Step 5: Thus b = a + c = 5, so the middle digit is 5. Step 6: When the digits are reversed, the number becomes 100c + 10b + a. Step 7: It is given that the reversed number is 99 more than the original, so (100c + 10b + a) − (100a + 10b + c) = 99. Step 8: Simplify the left side: (100c − 100a) + (a − c) = 99c − 99a = 99(c − a). Step 9: So 99(c − a) = 99, which implies c − a = 1. Step 10: We already have a + c = 5 and now c − a = 1. Adding the two equations gives 2c = 6, so c = 3, and then a = 2. Step 11: The digits are a = 2, b = 5, c = 3, giving the number 253.

Verification / Alternative check:
Check the conditions for 253. The sum of the digits is 2 + 5 + 3 = 10. The middle digit 5 is indeed equal to 2 + 3. When we reverse the digits, we get 352. The difference 352 − 253 is 99, exactly as stated in the problem. This confirms that 253 satisfies all the given conditions and is therefore the correct answer.

Why Other Options Are Wrong:
Option (b) 263: The sum of digits is 11, not 10.
Option (c) 273: The middle digit 7 is not equal to 2 + 3, and the reversal condition fails.
Option (d) 283: The digit relationships and the 99 difference condition are not satisfied.

Common Pitfalls:
Learners may forget that b must be exactly the sum of a and c or may mis-handle the place value expressions for the original and reversed numbers. It is also easy to make arithmetic errors when simplifying 99(c − a). Writing each equation carefully and double-checking arithmetic helps ensure the correct result.

Final Answer:
The three-digit number is 253.