Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #13
Grade 6 arithmeticPick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the total of all ages
Turn the overall average into a grand total: everyone's ages sum to 680.
Mean = sum ÷ count, so sum = mean × count. This is the Grade 6 definition of average rearranged.
6.SP.B.5Work BackwardsFind the girls' and boys' sums
Do the same for each kid group: the girls' ages sum to 300 and the boys' to 240.
Breaking the big group into two easier pieces keeps the arithmetic small and avoids any algebra.
6.SP.B.5Solve An Easier Related ProblemSubtract to get the adults' sum
Total minus the two kid sums leaves the adults: their ages sum to 140.
Same grand total, two different ways of counting — what's missing from the second way must be the adults' share.
The five adults' ages together add up to 140 years.
▸ Why?
Every member is in exactly one of the three groups — girls, boys, or adults — so the adults' total is what is left of the whole-camp age total after the girls' and boys' totals are removed.
▸ Why?
The 40 members split into girls, boys, and adults with nobody left out and nobody counted twice, so the three group age-totals add back to the total of all ages.
▸ Why?
Girls' total + boys' total + adults' total = grand total, so taking the two known totals away from the grand total leaves the adults' total, since subtracting undoes the adding that combined them.
▸ Why?
The grand total of all ages is 17 × 40 = 680 and the girls-and-boys total is 15 × 20 + 16 × 15 = 300 + 240 = 540, each found by multiplying an average by its count.
▸ Why?
An average is a group's total shared out equally among its members, so the total is exactly that average multiplied by how many members there are.
Divide to get their average
Divide that 140 among the 5 adults to reach their average age.
Back to the Grade 6 mean formula: average = sum ÷ count, with sum = 140 and count = 5.
6.SP.B.5Solve An Easier Related ProblemTotal 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.
- Find the total of all ages
- Find the girls' and boys' sums
- Subtract to get the adults' sum
- Divide to get their average
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