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, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's 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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Loki, Moe, and Nick each gave some of their money to Ott. Moe gave $\tfrac{1}{5}$ of his, Loki gave $\tfrac{1}{4}$ of his, and Nick gave $\tfrac{1}{3}$ 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?

Givens: Ott started with $\$0$; Moe gave Ott $\tfrac{1}{5}$ of Moe's money; Loki gave Ott $\tfrac{1}{4}$ of Loki's money; Nick gave Ott $\tfrac{1}{3}$ of Nick's money; The three gifts are equal in dollar amount; Answer choices: (A) $\tfrac{1}{10}$, (B) $\tfrac{1}{4}$, (C) $\tfrac{1}{3}$, (D) $\tfrac{2}{5}$, (E) $\tfrac{1}{2}$

Unknowns: Ott's share of the group's total money, as a fraction

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #6 Guess and Check

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.

Execute — Answer: B

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the shared gift.
Let $x$ be the number of dollars each friend gave to Ott.
Ott receives $x$ from Moe, $x$ from Loki, and $x$ from Nick, for a total of $3x$ dollars.

$$\text{Ott's money} = x + x + x = 3x$$

💡 When a problem says "the same amount each time," giving that amount a single letter usually unlocks the rest.

#4 Introduce a Variable 6.EE.B.7 Step 2

Back out each giver's starting money.
Moe gave $\tfrac{1}{5}$ of his money and the gift was $x$, so $\tfrac{1}{5} \cdot (\text{Moe's start}) = x$, meaning Moe started with $5x$.
Loki gave $\tfrac{1}{4}$ of his and got $x$, so Loki started with $4x$.
Nick gave $\tfrac{1}{3}$ of his and got $x$, so Nick started with $3x$.

$$\tfrac{1}{5}M = x \Rightarrow M = 5x;\quad \tfrac{1}{4}L = x \Rightarrow L = 4x;\quad \tfrac{1}{3}N = x \Rightarrow N = 3x$$

💡 "A fraction of my money equals $x$" reverses into "my money equals $x$ divided by that fraction" — multiply by the reciprocal.

#4 Introduce a Variable 6.EE.A.3 Step 3

Add up the group's total money.
The total stays the same before and after the gifts move around, so just add the starting balances: Moe + Loki + Nick + Ott (who started at $0$).

$$\text{Total} = 5x + 4x + 3x + 0 = 12x$$

💡 Money only moves between friends, so the group total is fixed — adding starting balances is the same as adding final balances.

#4 Introduce a Variable 6.RP.A.1 Step 4

Form the fraction Ott has now.
Ott now has $3x$ out of a group total of $12x$, so divide.

$$\dfrac{\text{Ott}}{\text{Total}} = \dfrac{3x}{12x} = \dfrac{3}{12} = \dfrac{1}{4} \;\Rightarrow\; \textbf{(B)}$$

💡 The $x$ cancels — exactly what we hoped, since the answer cannot depend on the actual dollar amount.

[1] #4 6.EE.A.2 Name the shared gift. Let $x$ be the number of dollars each friend gave to Ott.

[2] #4 6.EE.B.7 Back out each giver's starting money. Moe gave $\tfrac{1}{5}$ of his money and t

[3] #4 6.EE.A.3 Add up the group's total money. The total stays the same before and after the gi

[4] #4 6.RP.A.1 Form the fraction Ott has now. Ott now has $3x$ out of a group total of $12x$, s

Review

Reasonableness: 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 $\tfrac{15}{60} = \tfrac{1}{4}$, 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.

Alternative: Tool #6 (Guess and Check) skips the algebra entirely. Guess that each gift is $\$1$. Then Moe started with $\$5$, Loki $\$4$, Nick $\$3$, Ott $\$0$, total $\$12$. Ott now has $\$3$, and $\tfrac{3}{12} = \tfrac{1}{4}$ — answer (B). Because the problem's answer cannot depend on the gift size, this single test settles it.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the equal gift $x$ and writing Ott's total and each starting balance as multiples of $x$.)
6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form $px = q$ (Solving $\tfrac{1}{5}M = x$, $\tfrac{1}{4}L = x$, $\tfrac{1}{3}N = x$ for $M = 5x$, $L = 4x$, $N = 3x$.)
6.EE.A.3 Apply the properties of operations to generate equivalent expressions (Adding $5x + 4x + 3x = 12x$ to get the group's total.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Writing Ott's share as the ratio $\tfrac{3x}{12x} = \tfrac{1}{4}$ of the group's money.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

More like this