Problem 7 · 2003 AMC 8
Medium
Arithmetic & Operations
total-then-divide
Blake and Jenny each took four 100-point tests. Blake averaged 78 on the four tests. Compared with Blake, Jenny scored 10 points higher on the first test, 10 points lower on the second, and 20 points higher on each of the third and fourth. By how much does Jenny's average exceed Blake's on these four tests?
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Answer: A — 10 points.
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Hint 1 of 2
The number 78 is a decoy — the question only asks how far apart the two averages are, so work with the gaps, not the actual scores.
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Hint 2 of 2
The gap between two averages equals the average of the test-by-test gaps.
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Approach: average the differences, not the scores
- Ignore Blake's 78 — it cancels out of any difference. Just track how far ahead Jenny is each test: +10, −10, +20, +20.
- Add the gaps: 10 − 10 + 20 + 20 = +40 total points ahead over four tests. The difference of the averages is the average of those gaps: 40 ÷ 4 = 10 points.
- Why this works (and transfers): averages are linear — (Jenny's avg) − (Blake's avg) always equals the average of (Jenny − Blake) per test. Comparing two data sets? Subtract first, average second, and the shared baseline disappears.
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