Problem 7 · 2012 AMC 8
Easy
Arithmetic & Operations
average-budgetmin-with-max-on-other
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
Show answer
Answer: B — 92.
Show hints
Hint 1 of 2
Averages are slippery to push around — convert the goal into a fixed total. An average of 95 over 4 tests is the same as needing a certain number of total points.
Still stuck? Show hint 2 →
Hint 2 of 2
To make ONE score as small as possible, give the other score as much as it can take. The 4th test is the helper here, and it's capped at 100 — this "push the other to its limit" idea is the heart of every min/max problem.
Show solution
Approach: turn the average into a point budget, then max the helper
- Average 95 over 4 tests means a total of 4 × 95 = 380 points — a fixed budget is far easier to reason about than a moving average.
- She's already used 97 + 91 = 188, leaving 380 − 188 = 192 to split between tests 3 and 4.
- To make test 3 smallest, let test 4 grab the most it can: 100. Then test 3 = 192 − 100 = 92.
- The transferable move: to minimize one quantity in a fixed total, maximize the others (and vice versa). The cap (here, 100 points) is what makes the answer finite.
Mark:
· log in to save