AMC 8 · 2022 · #16

Grade 4 arithmeticalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

mean-median-mode-rangesystems-of-equationslinear-equations-two-var convert-to-algebraidentify-subproblems ↑ Prerequisites: mean-median-mode-rangelinear-equations-two-var

📏 Short solution 💡 3 insights

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Problem

Four numbers sit in a row. Call them a, b, c, d. We are told three two-number averages:

\frac{a+b}{2}

= 21,

\frac{b+c}{2}

= 26, and

\frac{c+d}{2}

= 30. The question asks for the average of the first and the last numbers,

\frac{a+d}{2}

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

Trying to solve for a, b, c, d individually is hopeless — three equations, four unknowns. But we do not need the individual values; we only need a+d. Tool #16 (Change Focus / Complement) is the key insight: inside the whole sum a+b+c+d, the pair a+d is exactly the complement of the middle pair b+c. If we know the whole and we know the middle, the outside is forced. Tool #7 (Identify Subproblems) breaks the work into three clean subproblems: (1) turn each given average into a pair sum by multiplying by 2, (2) build the whole-sum from the first and last pair sums, (3) subtract the middle pair sum, then halve to get the desired average. No algebra is needed — only the definition of average and four-operation arithmetic.

1STEP 1

Each given average is a pair's sum halved, so doubling it recovers that pair's sum.

a+b = 21 × 2 = 42, b+c = 26 × 2 = 52, c+d = 30 × 2 = 60

2STEP 2

Adding the outer pair sums a+b and c+d uses each number once, giving the whole a+b+c+d = 102.

(a+b) + (c+d) = 42 + 60 = 102, so a+b+c+d = 102

3STEP 3

The outer pair a+d is the complement of the middle: subtract b+c = 52 from the whole to get a+d = 50.

a+d = (a+b+c+d) - (b+c) = 102 - 52 = 50

4STEP 4

Halving the pair sum a+d turns it back into the requested average of the first and last numbers.

\frac{a+d}{2}

\frac{50}{2}

= 25 → (B)

Answer

The three given averages are 21, 26, 30 — increasing as we slide the window of two from the left end to the right end of the row. So the numbers themselves trend upward, and we should expect the average of the leftmost and rightmost numbers (a and d) to sit somewhere near the middle of that 21-to-30 range. The answer 25 lands right in that band — closer to the lower end, which fits because a (small end) drags the average down. A concrete check: a = 16, b = 26, c = 26, d = 34 satisfies all three given averages and gives

\frac{a+d}{2}

\frac{16+34}{2}

= 25. ✓

💡Key takeaway

This AMC 8 problem only needs Grade 4 multi-step arithmetic — add the outer pair sums, subtract the middle pair sum — that you already know!

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