AMC 10 · 2011 · #2
Grade 6 arithmeticPick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The goal is stated in terms of the final average, and we must recover a single missing score. Tool #11 (Work Backwards) runs the average formula in reverse: from the target average, find the total the six tests must reach, then subtract what she already has. Tool #7 (Identify Subproblems) splits the job into current-average, target-total, and needed-score. Tool #3 (Eliminate Possibilities) uses the answer choices to confirm that anything below 95 falls short.
Find the current average
The five scores add to 90+80+70+60+85=385, and 385÷5 gives a current average of 77.
An average is just the total shared equally across all the tests.
6.SP.B.5Identify SubproblemsSet the target average
At least 3 points above 77 means the new average must reach 77+3=80, and the smallest score comes from hitting it exactly.
The lowest passing score comes from just barely reaching the target, not overshooting it.
6.EE.B.5Work BackwardsWork backwards to the six-test total
The next test makes 6 tests, so an average of 80 needs the six scores to add to 80×6=480 points.
If the average must be 80 over six tests, the points must pile up to 80 six times over.
If the average must be a certain number over six tests, the points have to pile up to six times it.
▸ Why?
An average is a total shared over a count, so multiplying by the count undoes the sharing.
▸ Why?
That total is exactly the scores so far plus the missing one, so the gap names the answer.
Subtract to find the needed score
She already has 385, so the sixth score must close the gap: 480-385=95. Any smaller choice leaves the total under 480 — (E).
The missing score is exactly the leftover distance between what she has and the total she needs.
4.OA.A.3Eliminate PossibilitiesTo find one missing score for a target average, figure out the total all the tests must reach, then subtract the points you already have.
- Find the current average
- Set the target average
- Work backwards to the six-test total
- Subtract to find the needed score