A whole number is added to 100 and the same number is subtracted from 100. The…

Concept: For any number a and any number x, adding x to a and subtracting x from a produces two results whose sum is always 2a — the x terms are equal in size and opposite in sign, so they cancel regardless of the value of x.

Application: here a = 100, and the whole number that is added and subtracted is x.

1. Adding x to 100 gives the first resulting number: 100 + x.

2. Subtracting x from 100 gives the second resulting number: 100 - x.

3. Adding the two resulting numbers gives (100 + x) + (100 - x).

4. The +x and -x terms cancel, leaving 100 + 100 = 200.

Cross-check: choosing any whole number, say x = 37, gives the two resulting numbers 137 and 63, whose sum is 137 + 63 = 200 - confirming the result holds for any choice of x.

So the sum of the two resulting numbers is always 200, independent of which whole number is chosen.