Math CentralQuandaries & Queries


Question from Johan, a student:

Two numbers whose difference is 16 and whose sum is 120

Hi Johan,

I think you are expected to use algebra to solve this problem. If so here are two approaches.

  • Suppose the two numbers are $x$ and $y$ then you know that $x - y = 16$ and $x + y = 120.$ Solve these equations, perhaps by solving the first for $x$ and then substituting into the second.

  • Suppose the smaller of the two number is $z$ the the larger must be 16 more, that is $z + 16.$ The sum of the two numbers is 120. Solve for $z.$

I however wouldn't use algebra to solve this. I instead would ask "Find two numbers whose difference is zero and whose sum is 120." That is I want the two numbers to be the same and their sum is 120. What are they? Increase one of the numbers by 1 and decrease the other by 1. Their sum is still 120 but their difference is 2. Increase the large by 1 and decrease the smaller by 1. Their sum is still 120. What is their difference? How many times do you have to do this until their difference is 16?


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