Competition · AMC preparation · step 4 of 4

AMC 8 · 2004 · #22

Grade 7 probability
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
ratio-proportionprobability-basicfraction-arithmeticlinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: ratio-proportionfraction-arithmetic
📏 Medium solution 💡 3 insights
Problem
A party has only single women plus married couples (each married man comes with his wife). If you pick a woman at random, the chance she is single is 25\frac{2}{5}. What fraction of all the people in the room are married men?

Pick an answer.

(A)
$\frac13$
(B)
$\frac38$
(C)
$\frac25$
(D)
$\frac{5}{12}$
(E)
$\frac35$

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The problem hides every actual count and only gives a ratio, so Tool #4 (Introduce a Variable) is the right opening move: name the three groups and turn the words into equations. Two relations drop out immediately — "each married man has a wife here" gives M = W, and the probability 2/5 gives S/(S+W) = 2/5. With two equations and three unknowns we can express everything in one letter and read off the answer fraction. Tool #9 (Try a Simpler Case) is the natural backup: pick concrete head counts that match the 2/5 probability and just count.

1STEP 1

Name the three groups

Name the three groups: S single women, W wives, M married men — together they are everyone in the room.

total people = S + W + M
2STEP 2

Use the married-couple clue

Each married man brought his wife and each wife has a husband here, so they pair one-to-one: M = W.

M = W
3STEP 3

Turn the probability into an equation

A random woman is single with probability SS+W\frac{S}{S+W} = 25\frac{2}{5}; cross-multiply to get 3S = 2W.

S/(S + W) = 2/5 → 5S = 2(S+W) → 3S = 2W
4STEP 4

Write everything in one letter

Put all in terms of M: from M = W and 3S = 2W, substitute to get S = 23\frac{2}{3}M (and W = M).

W = M, S = 2/3M
5STEP 5

Find the married-men fraction

Married-men fraction MS+W+M\frac{M}{S+W+M} = M83M\frac{M}{\frac{8}{3}M} = 38\frac{3}{8}; the M cancels, so choice (B).

M/(2/3M + M + M) = M/8/3M = 3/8 → (B)
Answer
3/8
Plug in concrete numbers. Take M = 3 married men, so there are W = 3 wives and S = 2 single women — that gives 2 singles out of 5 women, matching the 25\frac{2}{5} probability. Total people = 2 + 3 + 3 = 8, and the married men are 3 of them, so the fraction is 38\frac{3}{8}. Matches (B). A quick sanity gut-check: roughly half the women are married, and married women come with husbands, so married men should be a noticeable chunk of the room — 38\frac{3}{8} (a bit less than half) fits.
💡Key takeaway

Married men match wives one-for-one, so the count of men equals the count of wives. Add that to the 25\frac{2}{5} probability and the answer 38\frac{3}{8} falls out — actual head counts never mattered.

  • Name the three groups
  • Use the married-couple clue
  • Turn the probability into an equation
  • Write everything in one letter
  • Find the married-men fraction

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