AMC 10 · 2007 · #12

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
linear-equations-two-varsystems-of-equations convert-to-algebra ↑ Prerequisites: linear-equations-one-var
📏 Medium solution 💡 2 insights
Problem
Tom's age right now is T years, and this same T is also the sum of the ages of his three children right now. Exactly N years ago, Tom's age was twice the sum of his three children's ages at that time. What is the ratio T/N?

Pick an answer.

(A)
2
(B)
3
(C)
4
(D)
5
(E)
6

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The problem already hands us two letters, T and N, so naming the remaining quantities in terms of them (Tool #4) is the natural move. The key that this tool unlocks is that 'N years ago' shifts each of the three children back by N, so their combined age drops by 3N, not by N. Once every age is written in T and N, the single sentence 'Tom's age then was twice their sum then' becomes one equation (Tool #13, Convert to Algebra), which we solve for the ratio T/N.

1STEP 1

Write every age in terms of T and N

Now Tom is T and the children sum to T. Rewind N years: Tom is T-N, and the three children each lose N, so their sum is T-3N.

Tom then=T-N, children's sum then=T-3N
2STEP 2

Turn the sentence into an equation

'Twice' makes the two sides equal: T-N=2(T-3N). Distributing the 2 on the right turns it into 2T-6N.

T-N=2(T-3N)=2T-6N
3STEP 3

Solve for the ratio T/N

Gather T on one side, N on the other: T-N=2T-6N gives 5N=T, so dividing by N leaves T/N=5, choice (D).

T-N=2T-6N → 5N=T → T/N=5
Answer
5
Pick numbers that fit T=5N. Let N=6, so T=30: Tom is 30 and his three children's ages sum to 30 now. Six years ago Tom was 24, and the children summed to 30-3(6)=12. Indeed 24=2 × 12, so the condition holds and T/N=30/6=5. Trying the tempting wrong answer that forgets the three children — using T-N=2(T-N) — would force T=N, i.e. Tom's whole age passed in N years, which is impossible, confirming the 3N drop is essential.
💡Key takeaway

When you rewind time in an age problem, every person loses those years, so three children lose three times as many — count all of them before writing the equation.

  • Write every age in terms of T and N
  • Turn the sentence into an equation
  • Solve for the ratio T/N

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