AMC 8 · 2007 · #7

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

mean-median-mode-rangemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

Problem

Five people in a room have an average age of 30. One of them, who is 18, walks out. What is the average age of the four people who stay?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Identify Subproblems) fits because averages do not subtract directly — you cannot just take 30 - 18. Instead the question splits into three small subproblems, each one a single Grade 6 step: (a) recover the original sum of ages from the original average, (b) subtract the leaver's age to get the new sum, (c) divide the new sum by the new count. No variable or equation is needed — the mean formula run forward and backward is enough.

1STEP 1

Recover the original total: sum = average × count, so five people averaging 30 give a total of 150 years.

original sum = 5 × 30 = 150 years

2STEP 2

Subtract the leaver: the 18-year-old walks out, so the total drops by exactly that, leaving 132 years.

new sum = 150 - 18 = 132 years

3STEP 3

Re-average: divide the new total by the new count of 4, giving 33.

new average =

\frac{132}{4}

= 33 → (D)

Answer

Check the direction first. The person who left was 18, well below the group average of 30, so pulling them out should raise the average — and 33 ＞ 30, which matches. Verify by running it the other way: if the 4 remaining people average 33, their total is 4 × 33 = 132. Add back the 18-year-old to get 132 + 18 = 150, and 150 / 5 = 30, the original average. The numbers close the loop.

💡Key takeaway

Averages do not subtract. To handle a "someone leaves" problem, go back to the total, take the leaver's age out, then divide by the new count.

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