AMC 8 · 2004 · #20

Grade 5 arithmetic

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

fraction-arithmeticratio-proportionlinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Short solution 💡 2 insights

📘 View easy version →

Problem

In a room,

\frac{2}{3}

of the people are seated and they fill

\frac{3}{4}

of the chairs. The other

\frac{1}{4}

of the chairs — that is, 6 chairs — are empty. How many people are in the room?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

How to solve

Strategy Work Backwards

The question asks for the number of people, but the only concrete number we are given is the 6 empty chairs, which is at the end of the chain. Tool #11 (Work Backwards) says: start from that known piece and undo the fractions one at a time. Tool #7 (Identify Subproblems) breaks the trip in two: first turn "6 empty chairs = $\frac{1}{4}$ of all chairs" into the total number of chairs, then turn "seated people = $\frac{2}{3}$ of all people" into the total number of people. Each subproblem is a one-step Grade 4-5 fraction question.

1STEP 1

The 6 empty chairs are the empty $\frac{1}{4}$ , so multiply by 4: the room has 24 chairs in all.

\frac{1}{4}

· (total chairs) = 6 → total chairs = 6 × 4 = 24

2STEP 2

The taken chairs are $\frac{3}{4}$ of 24, one person per chair, so 18 people are seated.

seated people =

\frac{3}{4}

× 24 = 18

3STEP 3

The 18 seated are $\frac{2}{3}$ of everyone; half of 18 is 9 for $\frac{1}{3}$ , so tripling gives 27 people total.

\frac{2}{3}

· (total people) = 18 →

\frac{1}{3}

· (total people) = 9 → total people = 3 × 9 = 27

4STEP 4

Read off the answer.

27 → (D)

Answer

Check by going forward. With 27 people, the seated group is

\frac{2}{3}

× 27 = 18 and the standing group is 27 - 18 = 9. With 24 chairs, the taken chairs are

\frac{3}{4}

× 24 = 18 (matches the 18 seated people) and the empty chairs are 24 - 18 = 6 (matches the problem). Eliminations: 12 and 18 are too small because

\frac{2}{3}

of 27 already equals 18. 24 is the chair count, not the people count — a classic trap answer. 36 would force

\frac{2}{3}

× 36 = 24 seated people, more than the 18 taken chairs allow. Only 27 is consistent end-to-end.

💡Key takeaway

When the only number you know is at the end of a fraction chain, work backwards: turn 6 empty chairs into 24 total chairs, 18 seated people, and finally 27 people in the room.

More like this