AMC 10 · 2007 · #10
Grade 7 arithmeticPick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The count of children is unknown, so Tool #4 (Introduce a Variable) names it n and lets every other quantity be written in terms of n. The trick that makes an average useful is to turn it back into a total: total age = average × number of people. Once both totals are written with n, Tool #13 (Convert to Algebra) turns the sentence 'father plus the rest equals the whole family' into one equation, which solves for n. Tool #6 (Guess and Check) then plugs the answer back to confirm the averages come out right.
Name the unknown, count the people
Let n be the number of children. The family is n + 2 people; the mother-and-children group drops the father, so it is n + 1.
Giving the unknown a name lets every headcount in the problem be written as a short expression in n.
6.EE.B.6Introduce A VariableTurn each average into a total
Total = average × count, so the family total is 20(n + 2) and the mother-and-children total is 16(n + 1).
Multiplying an average back by the number of people recovers the total you started from.
Multiplying an average back by the number of people recovers the total you started from.
▸ Why?
An average is a total shared over a count, so multiplying by that count undoes the sharing.
▸ Why?
The family total is exactly its parts added, so it can be built either way and both builds agree.
Split the family total two ways
The family is the father plus that group, so the two ways of writing the family total must be equal — one equation in n.
One whole total can be built either all at once or by adding its parts, and both builds must match.
7.EE.B.4Convert To AlgebraSolve the equation
Expand: 20n + 40 = 16n + 64, so 4n = 24 and n = 6 — six children, choice (E).
Collect the n terms on one side and the plain numbers on the other, then undo the multiplication.
6.EE.B.7Convert To AlgebraTurn every average back into a total by multiplying by the number of people, then match the whole against the sum of its parts to solve for the unknown.
- Name the unknown, count the people
- Turn each average into a total
- Split the family total two ways
- Solve the equation