Final answer:
By creating equations based on the given criteria for the number's digits and then testing each provided option, the correct number that satisfies all conditions is found to be 253.
Step-by-step explanation:
To solve this mathematical problem, we have a number that consists of three digits whose sum is 10. The middle digit is equal to the sum of the other two digits, and when the digits of the number are reversed, it is increased by 99. Let's call these three digits a, b, and c respectively, so our number is expressed as abc, where a and c are the outer digits and b is the middle digit. The conditions given can be expressed as the following equations:
- a + b + c = 10
- b = a + c
- 100c + 10b + a = 100a + 10b + c + 99
From equation 2, b = a + c, and substituting this and a + c = 10 - b (from equation 1) into equation 3, we simplify to find the value of a and c. We get:
100c + 10(a + c) + a = 100a + 10(a + c) + c + 99
Solving the simplified equation gives us 99c - 99a = 99, which simplifies further to c - a = 1. Therefore, digit c is one more than digit a. Checking the given options, we can now determine which one meets all the criteria specified in the problem.
Test each option to see if it satisfies all the requirements:
- (a) 145 does not meet the criteria since 1+4+5 = 10, but 4 is not equal to 1+5.
- (b) 253 meets all criteria: 2+5+3 = 10, 5 = 2+3, and when reversed we get 352, which is indeed 253 + 99.
- (c) 370 does not meet since the sum of the digits is not 10.
- (d) 352 does not meet the requirement that the middle digit equals the sum of the other two.
Based on our methodical elimination, the correct answer is option (b) 253.