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 is making a batch of cookies and needs $2\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2013 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.

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Toolkit + CCSS Solution

Understand

Restated: 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 the $\frac{1}{4}$-cup measure to reach exactly $2\frac{1}{2}$ cups.

Givens: The total sugar needed is $2\frac{1}{2}$ cups; Each fill of the measuring cup provides exactly $\frac{1}{4}$ cup; Answer choices: (A) $8$, (B) $10$, (C) $12$, (D) $16$, (E) $20$

Unknowns: The number of $\frac{1}{4}$-cup fills that add up to $2\frac{1}{2}$ cups

Understand

Givens: The total sugar needed is $2\frac{1}{2}$ cups; Each fill of the measuring cup provides exactly $\frac{1}{4}$ cup; Answer choices: (A) $8$, (B) $10$, (C) $12$, (D) $16$, (E) $20$

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

Tracking the units settles what operation to do: the total sugar is in cups, each fill is in cups-per-fill, and cups $\div$ cups-per-fill gives a plain count of fills. So the question is really $2\frac{1}{2} \div \frac{1}{4}$ — "how many $\frac{1}{4}$-cups fit inside $2\frac{1}{2}$ cups." Tool #7 (Identify Subproblems) makes that count easy by splitting $2\frac{1}{2}$ 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.

Execute — Answer: B

#8 Analyze the Units 5.NF.B.7 Step 1

Set up the count as a division

Each fill adds $\frac{1}{4}$ cup, and the amounts pile up equally, so the number of fills is the total sugar divided by the size of one fill.
Watching the units confirms it: cups $\div$ $\frac{\text{cups}}{\text{fill}}$ leaves fills.
So the answer is however many $\frac{1}{4}$-cups fit inside $2\frac{1}{2}$ cups.

$$\text{number of fills} = 2\frac{1}{2} \div \frac{1}{4}$$

💡 Dividing the total by the size of one scoop tells you how many scoops it takes.

#7 Identify Subproblems 4.NF.B.3 Step 2

Break the total into quarter-cups

Count in $\frac{1}{4}$-cups.
Each whole cup is four $\frac{1}{4}$-cups, so the $2$ whole cups are $2 \times 4 = 8$ of them.
The leftover $\frac{1}{2}$ cup is $\frac{2}{4}$, which is $2$ more quarter-cups.
Altogether $2\frac{1}{2}$ cups is made of $8 + 2 = 10$ quarter-cups.

$$2\frac{1}{2} = \frac{10}{4} = \underbrace{\tfrac14 + \tfrac14 + \cdots + \tfrac14}_{10\ \text{times}}$$

💡 A whole cup holds four quarter-cups, so counting in quarters just means counting the pieces.

#3 Eliminate Possibilities 5.NF.B.7 Step 3

Read off the number of fills

Since $2\frac{1}{2}$ cups is exactly $10$ quarter-cups, filling the $\frac{1}{4}$-cup measure $10$ times gives the right amount: $2\frac{1}{2} \div \frac{1}{4} = 10$.
The trap answer $8$ counts only the two whole cups and drops the half cup, so it is too small.
The correct count is $\textbf{(B)}\ 10$.

$$2\frac{1}{2} \div \frac{1}{4} = \frac{10}{4} \div \frac{1}{4} = 10 \;\Rightarrow\; \textbf{(B)}$$

💡 If ten quarter-cups make the total, then ten fills is the answer.

[1] #8 5.NF.B.7 Each fill adds $\frac{1}{4}$ cup, and the amounts pile up equally, so the number

[2] #7 4.NF.B.3 Count in $\frac{1}{4}$-cups. Each whole cup is four $\frac{1}{4}$-cups, so the $

[3] #3 5.NF.B.7 Since $2\frac{1}{2}$ cups is exactly $10$ quarter-cups, filling the $\frac{1}{4}

Review

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

Alternative: Use decimals instead of fractions. The measuring cup holds $0.25$ cup and Alice needs $2.5$ cups, so the number of fills is $2.5 \div 0.25 = 10$. This matches the fraction count and confirms choice (B).

CCSS standards used (min grade 5)

5.NF.B.7 Divide involving unit fractions in real-world problems (Reading the problem as $2\frac{1}{2} \div \frac{1}{4}$ — how many $\frac{1}{4}$-cups fit in the total — and computing the quotient $10$.)
4.NF.B.3 Understand a fraction as a sum of unit fractions (Rewriting $2\frac{1}{2}$ cups as ten $\frac{1}{4}$-cups ($8$ from the whole cups plus $2$ from the half cup).)

⭐ To find how many small scoops fill a total, divide the total by one scoop — here $2\frac{1}{2}$ cups is exactly ten quarter-cup scoops.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Set up the count as a division

Break the total into quarter-cups

Read off the number of fills

Review

CCSS standards used (min grade 5)

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