AMC 10 · 2015 · #3
Grade 6 arithmeticPick an answer.
An average hides the total, so Tool #7 (Identify Subproblems) splits the work into clean pieces: first turn each average back into a sum of points, once for the 14 tests and once for all 15. Tool #11 (Work Backwards) then recovers the single missing number — Payton's score is whatever was added to the old total to make the new total, so subtract the two sums.
Total of the 14 graded tests
The first average hides a total of 1120.
An average is just the total shared out evenly, so multiplying it back by the count rebuilds the total.
6.SP.B.5Identify SubproblemsTotal of all 15 tests
The second hides a total of 1215.
The same total = average × count rule turns the new average into the new full total.
5.NBT.B.5Identify SubproblemsSubtract to find Payton's score
The gap is 95, choice (E).
Adding one score raises the total by exactly that score, so the rise in the total is the score itself.
Adding one score raises the total by exactly that score, so the rise in the total is the score itself.
▸ Why?
An average is a total shared over a count, so each average can be turned back into its own total.
▸ Why?
The two totals share the same fourteen scores, so subtracting leaves only the newly added one.
Turn each average back into a total by multiplying by the count, then the score you added is just the jump in the total.
- Total of the 14 graded tests
- Total of all 15 tests
- Subtract to find Payton's score