AMC 10 · 2005 · #3

Grade 5 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

fraction-multiplicationfraction-arithmetic identify-subproblemscomplementary-counting ↑ Prerequisites: fraction-arithmetic

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Problem

One gallon of paint is used up over three days. On day 1,

\frac{1}{3}

of the paint is used. On day 2,

\frac{1}{3}

of whatever paint is left is used. Find what fraction of the original gallon is still available on day 3.

Pick an answer.

(A)

$\frac{1}{10}$

(B)

$\frac{1}{9}$

(C)

$\frac{1}{3}$

(D)

$\frac{4}{9}$

(E)

$\frac{5}{9}$

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The paint changes one day at a time, so Tool #7 (Identify Subproblems) handles it as two small steps: find what is left after day 1, then find what day 2 removes. Tool #16 (Change Focus / Count the Complement) keeps the eye on what is LEFT rather than what is used — the whole trap of the problem, since the paint USED totals 5/9, which is choice (E). Tool #3 (Eliminate Possibilities) then rules out the copy-cat traps: 1/9 if you wrongly take 1/3 of 1/3, and 1/3 if you stop after day 1.

1STEP 1

Paint left after day 1

Write the whole gallon as thirds and take day 1's share away: $1-\frac{1}{3}=\frac{3}{3}-\frac{1}{3}=$ $\frac{2}{3}$ is left when day 2 begins.

1-1/3=3/3-1/3=2/3

2STEP 2

Paint used on day 2

Day 2 takes a third of what is left, not of the whole gallon, and "of" means multiply: $\frac{1}{3}\times\frac{2}{3}=$ $\frac{2}{9}$ .

1/3×2/3=2/9

3STEP 3

Paint available on day 3

Subtract what day 2 used from what day 2 started with: $\frac{2}{3}-\frac{2}{9}=\frac{6}{9}-\frac{2}{9}=$ $\frac{4}{9}$ , choice (D).

2/3-2/9=6/9-2/9=4/9 → (D)

Answer

4/9

Check with the complement: used =5/9 and left =4/9, and 5/9+4/9=9/9=1 whole gallon — the two parts fill exactly one gallon, so the split is consistent. The leftover 4/9≈0.44 is a bit less than half, which fits: after taking a third, then a third of the rest, more than half the gallon is gone but not much more. Choice (E) 5/9 is the amount used, (C) 1/3 stops after day 1, and (B) 1/9 comes from multiplying 1/3×1/3 instead of 1/3×2/3.

💡Key takeaway

"A third of what's left" means multiply by the leftover, not the whole — and since the question asks what's still there, track what stays, not what's gone.

Paint left after day 1
Paint used on day 2
Paint available on day 3

Paint left after day 1

Paint used on day 2

Paint available on day 3

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