AMC 10 · 2020 · #2

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

mean-median-mode-rangelinear-equations-one-varwork-backwards work-backwardsidentify-subproblems ↑ Prerequisites: mean-median-mode-range

📏 Short solution 💡 2 insights

Problem

Five numbers — 3, 5, 7, a, and b — have an average of 15. Find the average of just a and b.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

Tool #11 (Work Backwards) is the spine: the given is the end product (the average 15), and we need the missing pieces (a and b). Undo the average by multiplying back: total sum = 15 × 5 = 75. Tool #7 (Subproblems) splits the work cleanly — first the sum of all five, then the sum of a + b, then the average. Tool #3 (Eliminate) is a sanity check: since the known three (3, 5, 7) are all below the average of 15, the missing two must be substantially above 15 — so (A) 0 and (B) 15 are impossible.

1STEP 1

Undo the average. Five numbers with mean 15 have a total sum of 5 × 15 = 75.

sum of all five = 5 × 15 = 75

2STEP 2

Subtract the three known numbers. Since 3 + 5 + 7 = 15, we get a + b = 75 - 15 = 60.

3 + 5 + 7 = 15, a + b = 75 - 15 = 60

3STEP 3

Average just a and b by halving their sum: $\frac{a + b}{2}$ = $\frac{60}{2}$ = 30.

\frac{a + b}{2}

\frac{60}{2}

= 30 → (C)

4STEP 4

Elimination check: 3, 5, 7 sit 30 below the mean 15, so a and b average 15 above it, i.e. 30 — confirms (C).

(15-3) + (15-5) + (15-7) = 30 → avg of a, b = 15 + 15 = 30

Answer

Sanity: if a + b = 60 then the five-number sum is 3 + 5 + 7 + 60 = 75, and

\frac{75}{5}

= 15. The original condition holds. Magnitude-wise, 30 is larger than 15 which makes sense because a and b have to push the low values 3, 5, 7 up to a mean of 15.

💡Key takeaway

This AMC 10 problem only needs Grade 6 "the mean is the total split evenly" you already know — multiply back to 75, take out 3 + 5 + 7 = 15, and the missing pair sums to 60, so their average is 30.

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