Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #12
Grade 6 rate-ratioPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Turn percents into counts
Convert each percent to a count of correct answers by multiplying it (as a decimal) by that test's number of questions.
"Percent of a quantity as a rate per 100" is the Grade 6 way to read "70% of 10" as 7.
6.RP.A.3Solve An Easier Related ProblemTotal the correct answers
Add the three correct counts to get 50 right out of the 60 total questions.
Once the percents are gone, this is just whole-number addition — the Grade 5 fluency standard.
5.NBT.B.5Make A Systematic ListSimplify the ratio
Write the score as total correct over total questions and simplify: = .
50/60 and 5/6 are the same ratio written with different sized parts — the Grade 6 ratio idea.
On the merged 60-question test, Antonette's overall score is the pooled correct answers over the pooled questions — 50 out of 60 — found by adding the counts, not by averaging the three percents.
▸ Why?
The 50 correct answers on the merged test are the three tests' correct counts — 7, 16, 27 — combined with none dropped and none double-counted.
▸ Why?
Merging the tests lays their correct answers side by side with no gaps or overlaps, so the separate counts add back to one whole count.
▸ Why?
Each test's count is that test's percent of its own questions — 70% of 10 is 7, 80% of 20 is 16, 90% of 30 is 27 — because a percent is a count per hundred, so p% of a number is p hundredths of it.
▸ Why?
The 60 questions on the merged test are the three tests' lengths — 10, 20, 30 — combined the same way, with no question left out or counted twice; because these parts differ in size, the 30-question test weighs more than the 10-question one, which is why a plain average of the percents is wrong.
Convert to a percent
Divide 5 by 6 to get about 83.3%, then pick the closest choice.
83.33% rounds to 83%, which is choice (D). Choice (C) 80% is the trap from naively averaging 70, 80, 90.
6.RP.A.3Solve An Easier Related ProblemWhen 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.
- Turn percents into counts
- Total the correct answers
- Simplify the ratio
- Convert to a percent
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