Problem 25 · 2002 AMC 8
Stretch
Algebra & Patterns
substitution
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?
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Answer: B — 1/4.
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Hint 1 of 2
The fractions 1/5, 1/4, 1/3 look mismatched, but the problem hands you a gift: all three gifts are the *same dollar amount*. Name that shared amount — say $1 — and everything else falls out.
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Hint 2 of 2
If $1 is one-fifth of Moe's money, Moe had $5; likewise Loki $4, Nick $3. And notice money only moves *around* — the group total never changes.
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Approach: set the equal gift to a convenient $1
- Anchor on the equal gift: let each be $1. Then $1 = ⅕ of Moe → $5, = ¼ of Loki → $4, = ⅓ of Nick → $3.
- Giving money away doesn't create or destroy any: the group total stays $5 + $4 + $3 = $12. Ott now holds the three $1 gifts = $3.
- Ott's share = 3/12 = 1/4.
- *Two ideas worth keeping:* (1) when several quantities share a common value, set *that* value to a friendly number and back out the rest; (2) money passed within a group conserves the total — so a "fraction of the whole" question only needs the unchanged grand total as its denominator.
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