AMC 10 · 2003 · #10

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

ratio-proportionfraction-multiplicationfraction-arithmetic complementary-countingeasier-related-problem ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 2 insights

Problem

A pile of candy is supposed to be split among Al, Bert, and Carl in the ratio 3:2:1. They show up one at a time, and each one wrongly believes the pile in front of him is the whole untouched prize, so each takes his ratio share of whatever he happens to find. Find the fraction of the original pile that is still sitting there when all three have gone.

Pick an answer.

(A)

$\frac{1}{18}$

(B)

$\frac{1}{6}$

(C)

$\frac{2}{9}$

(D)

$\frac{5}{18}$

(E)

$\frac{5}{12}$

How to solve

Strategy Change Focus / Count the Complement

Adding up the three amounts taken is awkward, because each amount depends on how much the previous person already removed. Tool #16 (Change Focus) flips the bookkeeping: instead of tracking what leaves, track what stays. Each visit simply multiplies the leftover pile by a fixed survival fraction, and multiplying three fixed numbers is far easier than chaining three subtractions. Tool #4 (Introduce a Variable) sets the original pile to 1, so "amount left" and "fraction left" become the same number. Tool #9 (Solve an Easier Related Problem) gives a clean check: rerun the whole story with a convenient whole number of candies, where every step is a plain count with no fractions at all.

1STEP 1

Turn the ratio into shares

The ratio makes six parts, so the shares are 1/2, 1/3 and 1/6 of a full pile.

3+2+1=6 → Al 3/6=1/2, Bert 2/6=1/3, Carl 1/6

2STEP 2

Track what survives each visit

Each visit scales the pile, leaving 1/2, 2/3 and 5/6 of whatever was found.

Al leaves 1/2, Bert leaves 2/3, Carl leaves 5/6 of the pile he finds

3STEP 3

Walk the pile through in order

Walking through in order takes the pile from 1 to 1/2 to 1/3 to 5/18.

1 → 1/2 → 1/2-1/6=1/3 → 1/3-1/18=5/18

4STEP 4

Multiply the survival factors

Multiplying the three factors gives the same 5/18 in any order, choice (D).

1·1/2·2/3·5/6=10/36=5/18 → (D)

Answer

5/18

The answer must sit strictly between 0 and 1, and it must be less than 1/3, since after Al and Bert only 1/3 of the pile is left and Carl still takes some of that. That upper bound alone rules out (E) 5/12≈ 0.42, and 5/18≈ 0.28 passes. It must also be more than 2/9=4/18, because Carl only removes a sixth of the 1/3 that remained, not a whole 1/9 of the pile, so (C) is too small and the very small choices (A) and (B) are far too small. Counting up the amounts taken confirms the total: 1/2+1/6+1/18=9/18+3/18+1/18=13/18, and 13/18+5/18=1, exactly the whole pile.

💡Key takeaway

When people take turns removing a fraction of whatever is left, follow the part that survives each turn and multiply those survivors together.

Turn the ratio into shares
Track what survives each visit
Walk the pile through in order
Multiply the survival factors

Turn the ratio into shares

Track what survives each visit

Walk the pile through in order

Multiply the survival factors

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