Problem 12 · 2006 AMC 8
Easy
Fractions, Decimals & Percents
weighted-percentage
Antonette gets 70% on a 10-problem test, 80% on a 20-problem test and 90% on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score?
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Answer: D — 83%.
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Hint 1 of 2
You can NOT just average 70, 80, 90 — the tests have different sizes, so the bigger tests should count for more. Go back to actual problems right and wrong.
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Hint 2 of 2
A percentage is a fraction in disguise. Turn each percent into a count of correct problems, pool them, then form one big fraction: total right ÷ total problems. This is a weighted average.
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Approach: count actual correct problems, then one overall fraction
- Convert each percent to a count: 70% of 10 = 7, 80% of 20 = 16, 90% of 30 = 27.
- Pool them: 7 + 16 + 27 = 50 correct out of 60 total.
- 50 ÷ 60 ≈ 0.833 ⇒ closest to 83%.
- Why not 80%? The naive average of 70, 80, 90 is 80 — that's answer choice C, the trap. It would only be right if the tests were the same size. Because the 90% test is the biggest (30 problems), it pulls the real score above 80. Always weight by size.
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