AMC 10 · 2013 · #2

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

fraction-arithmeticfraction-multiplicationratio-proportion identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 1 insight

Problem

Alice needs

2\frac{1}{2}

cups of sugar, but her measuring cup holds only

\frac{1}{4}

cup. Find how many times she must fill that

\frac{1}{4}

-cup measure to reach exactly

2\frac{1}{2}

cups.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

Tracking the units settles what operation to do: the total sugar is in cups, each fill is in cups-per-fill, and cups ÷ cups-per-fill gives a plain count of fills. So the question is really 2 1/2 ÷ 1/4 — "how many 1/4-cups fit inside 2 1/2 cups." Tool #7 (Identify Subproblems) makes that count easy by splitting 2 1/2 into the two whole cups and the leftover half. Tool #3 (Eliminate Possibilities) catches the trap answer 8, which counts only the two whole cups and forgets the half.

1STEP 1

Set up the count as a division

Each fill adds the same $\frac{1}{4}$ cup, and cups ÷ cups-per-fill leaves fills, so the count is the total divided by one fill.

number of fills = 2 1/2 ÷ 1/4

2STEP 2

Break the total into quarter-cups

One cup is four $\frac{1}{4}$ -cups, so 2 cups give 8, and the leftover $\frac{1}{2}$ cup adds 2 more: 10 quarter-cups.

2 1/2 = 10/4 = 1/4 + 1/4 + … + 1/4₁₀ times

3STEP 3

Read off the number of fills

So filling the measure 10 times lands exactly on $2\frac{1}{2}$ cups; the trap 8 drops the half cup, so (B) is right.

2 1/2 ÷ 1/4 = 10/4 ÷ 1/4 = 10 → (B)

Answer

The total 2 1/2 cups is a bit more than 2 cups, and 2 cups alone already need 2 × 4 = 8 fills, so the answer must be a little above 8 — and 10 is. It also lands exactly on the total: 10 × 1/4 = 10/4 = 2 1/2 cups, no rounding needed. Choice 8 (forgetting the half cup) is too small, and 20 (as if the cup held 1/8) is far too big, so 10 is the sensible fit.

💡Key takeaway

To find how many small scoops fill a total, divide the total by one scoop — here 2 1/2 cups is exactly ten quarter-cup scoops.

Set up the count as a division
Break the total into quarter-cups
Read off the number of fills

Set up the count as a division

Break the total into quarter-cups

Read off the number of fills

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