Problem 17 · 1985 AJHSME
Hard
Ratios, Rates & Proportions
total-difference
If your average score on your first six mathematics tests was 84 and your average score on your first seven mathematics tests was 85, then your score on the seventh test was
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Answer: D — 91.
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Hint 1 of 2
An average is a disguised total: average × count = sum. The seventh test is the only difference between the two groups, so it must be the gap between the two totals.
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Hint 2 of 2
Turn each average into a running total (7 × 85 and 6 × 84), then subtract — what's left over is the one extra test.
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Approach: difference of totals
- Recover the sums: first 7 tests total 7 × 85 = 595; first 6 tests total 6 × 84 = 504.
- The seventh test is the difference: 595 − 504 = 91.
- Intuition: adding the 7th test pulled the average UP by 1 point across all 7 tests. So the 7th score had to be the old average 84 plus enough to lift all seven by 1 — that's 84 + 7 = 91. (Quick check: 84 + 7×1 = 91 ✓.)
Mark:
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