AMC 8 · 2003 · #7

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

mean-median-mode-rangemulti-digit-arithmeticset-partition identify-subproblemspattern-recognition ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

Problem

Blake and Jenny each take four 100-point tests. Blake averages 78. Jenny scores +10, -10, +20, +20 relative to Blake on tests 1, 2, 3, 4. Find Jenny's average minus Blake's average.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question only asks for the difference of two averages, not either average by itself, which is why Blake's 78 never has to be used. Tool #7 (Identify Subproblems) splits the work into two clean pieces: (1) add the four per-test score differences to get the total difference, then (2) divide that total by the number of tests to get the difference of the averages. Tool #11 (Work Backwards) here just means reading the mean formula in reverse — average = total ÷ count, so difference of averages = difference of totals ÷ count.

1STEP 1

Subproblem 1: read the four per-test gaps (Jenny minus Blake) straight off the problem: +10, -10, +20, +20.

D₁ = +10, D₂ = -10, D₃ = +20, D₄ = +20

2STEP 2

Add the four signed gaps: +10 and -10 cancel, leaving 20 + 20 = 40 as the total point gap.

D₁ + D₂ + D₃ + D₄ = 10 + (-10) + 20 + 20 = 40

3STEP 3

Subproblem 2: same test count, so the average gap is the total gap over 4 — Jenny's average minus Blake's is $\frac{40}{4}$ .

Jenny's avg - Blake's avg =

\frac{Jenny's total - Blake's total}{4}

\frac{40}{4}

4STEP 4

Divide to finish: $\frac{40}{4}$ = 10, choice (A).

\frac{40}{4}

= 10 → (A)

Answer

Sanity-check with concrete scores. Pretend Blake scored 78 on every test (total 312, average 78). Then Jenny scored 88, 68, 98, 98, summing to 352 with average 88. Difference of averages: 88 - 78 = 10, matching (A). Trap choices map to common mistakes: (E) 40 stops at the total difference and forgets to divide by 4, (C) 20 averages the four differences but mistakenly drops the negative sign on D₂, and (B) 15 / (D) 25 are off-pattern distractors that test whether students just guess.

💡Key takeaway

When a question asks for the difference of two averages over the same number of tests, you don't need either average. Add the per-test gaps and divide by the count — here, 40 ÷ 4 = 10.

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