AMC 8 · 2002 · #25

Grade 6 rate-ratioalgebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Integrality / Divisibility Screen

ratio-proportionfraction-arithmeticfraction-multiplicationsystems-of-equations convert-to-algebraidentify-subproblemspattern-recognition ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Medium solution 💡 3 insights

Problem

Loki, Moe, and Nick each gave some of their money to Ott. Moe gave

\frac{1}{5}

of his, Loki gave

\frac{1}{4}

of his, and Nick gave

\frac{1}{3}

of his — and all three gifts were the same number of dollars. Ott started with nothing. What fraction of the four friends' combined money does Ott now have?

Pick an answer.

(A)

$\frac{1}{10}$

(B)

$\frac{1}{4}$

(C)

$\frac{1}{3}$

(D)

$\frac{2}{5}$

(E)

$\frac{1}{2}$

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The problem never gives a dollar amount, only fractions and the rule "each gift is equal." That equal-gift dollar value is the hidden link, so Tool #4 (Introduce a Variable) — call it x — turns every starting balance into a multiple of x. Once Moe has 5x, Loki has 4x, and Nick has 3x, the answer is a ratio of dollar counts. Tool #6 (Guess and Check) gives a quick concrete sanity check: pick x = $5 and confirm the same fraction appears.

1STEP 1

Let x be the equal gift each friend hands Ott, so Ott collects it three times for 3x.

Ott's money = x + x + x = 3x

2STEP 2

Reverse each fraction: Moe started with 5x, Loki with 4x, and Nick with 3x.

\frac{1}{5}

M = x → M = 5x;

\frac{1}{4}

L = x → L = 4x;

\frac{1}{3}

N = x → N = 3x

3STEP 3

Nothing is created or lost, so add the starting balances: the group holds 12x.

Total = 5x + 4x + 3x + 0 = 12x

4STEP 4

Ott's 3x over the group's 12x reduces to $\frac{1}{4}$ — choice (B).

Ott/Total = 3x/12x =

\frac{3}{12}

\frac{1}{4}

→ (B)

Answer

\frac{1}{4}

Pick a concrete value to double-check: let x =

5. Then Moe started with

25, Loki with

20, Nick with

15, and Ott with

0, for a group total of

60. Each friend hands

5 to Ott, so Ott ends with

15. The fraction is

\frac{15}{60}

\frac{1}{4}

, matching (B). The size also makes sense: Ott has one gift from each of three friends, so he holds roughly a quarter of the four-person pot.

💡Key takeaway

When three different fractions all give the same amount, name that amount x and let each starting balance fall out — the unknown x cancels at the end, so the answer is a ratio of plain counts.

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