AMC 8 · 1999 · #13

Grade 6 arithmetic

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

mean-median-mode-rangemulti-digit-arithmeticlinear-equations-one-var identify-subproblems ↑ Prerequisites: multi-digit-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

Problem

A computer science camp has 40 members whose average age is 17. The members split into 20 girls (average age 15), 15 boys (average age 16), and 5 adults. Find the average age of the adults.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Find an Invariant

The total sum of all 40 ages is the invariant here (Tool #11): you can compute it two ways — directly from the overall average (17 × 40), or by adding the three subgroup sums (girls + boys + adults). Both must give the same number, which pins down the adults' total. Tool #9 (Solve an Easier Problem) splits the work into three small "average × count" calculations instead of one big algebra setup.

1STEP 1

Turn the overall average into a grand total: everyone's ages sum to 680.

Total sum = 17 × 40 = 680

2STEP 2

Do the same for each kid group: the girls' ages sum to 300 and the boys' to 240.

Girls' sum = 15 × 20 = 300, Boys' sum = 16 × 15 = 240

3STEP 3

Total minus the two kid sums leaves the adults: their ages sum to 140.

Adults' sum = 680 - 300 - 240 = 140

4STEP 4

Divide that 140 among the 5 adults to reach their average age.

Adults' average =

\frac{140}{5}

= 28 → (C)

Answer

Plug 28 back in and rebuild the overall average. Total ages = 20(15) + 15(16) + 5(28) = 300 + 240 + 140 = 680. Divide by 40 members: 680 ÷ 40 = 17, matching the given overall average. Also, the adults' average (28) is well above the kids' averages (15 and 16), which makes sense — only 5 adults have to pull the camp average up from the mid-teens to 17, so each adult must be much older than the kids.

💡Key takeaway

Total age stays the same no matter how you split the camp — turn each "average" into a sum, add the pieces, and the missing piece pops right out.

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