AMC 8 · 2006 · #12

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentagefraction-arithmeticmulti-digit-arithmeticratio-proportion identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

Antonette scores 70% on a 10-question test, 80% on a 20-question test, and 90% on a 30-question test. If all three tests are merged into one 60-question test, which choice is closest to her overall percent correct?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The trap is averaging 70, 80, 90 to get 80 (choice C). That ignores test size. Tool #9 (Solve an Easier Problem) reframes the question: instead of mixing percents, count actual questions answered correctly on each test — those are concrete numbers we can add. Tool #2 (Make a Systematic List) then walks through the three tests in order, recording correct count and total count for each, so the combined total is just two sums to divide. No algebra needed.

1STEP 1

Convert each percent to a count of correct answers by multiplying it (as a decimal) by that test's number of questions.

0.70 × 10 = 7, 0.80 × 20 = 16, 0.90 × 30 = 27

2STEP 2

Add the three correct counts to get 50 right out of the 60 total questions.

Total correct = 7 + 16 + 27 = 50, Total questions = 10 + 20 + 30 = 60

3STEP 3

Write the score as total correct over total questions and simplify: $\frac{50}{60}$ = $\frac{5}{6}$ .

\frac{50}{60}

\frac{5}{6}

4STEP 4

Divide 5 by 6 to get about 83.3%, then pick the closest choice.

\frac{5}{6}

= 0.83 = 83.3% → (D) 83

Answer

The bigger tests (the 20-question and 30-question ones) score higher than 70%, and the 30-question test — the heaviest — scores the highest at 90%. So the overall average should pull toward 90%, landing above the naive mean of 80%. Our answer 83.3% sits between 80% and 90% and is closer to 80%, which makes sense because the 30-question test is only half of the 60-question total. Choice (D) 83 fits; choice (E) 87 would mean the 90% test dominated more than it really does.

💡Key takeaway

When tests have different sizes, you can't just average the percents. Turn each percent into a count of correct answers, add them up, and divide by the total number of questions — then the right percent falls out.

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