Diagnosing tricky questions

I’m curious how @orion would categorize the following question from ETS power prep practice test 1 in the achievable diagnostic framework.

If x and y are the tens and the units digit, respectively, of the product 725,278×67,066, what is the value of x+y ?

  1. 12
  2. 10
  3. 8
  4. 6
  5. 4
1 Like

Hi @James1! Orion here, to answer your question.

This question is not difficult because of the math (which is simply multiplication and addition), but because of the size of the numbers involved. They won’t fit in the on-screen calculator! When this occurs, you probably have a counting problem on your hands.

My go-to technique for solving counting problems is shrinking. That is, I create an analogous problem with smaller numbers, solve that problem instead, and generalize to the original question. In this case, since the question is only apparently concerned with the tens and units (i.e., ones) digits of the product, I can limit myself to the tens and units digits of the multipliers. This shrinks the problem from 725,278 x 67,066 to 78 x 66, which is solvable using the on-screen calculator (i.e, 5,148). The tens and units digits of this product are equivalent to the tens and units digits of the original product. And since 4 + 8 is 12, we’d go with answer choice A.

Hope this helps!