AMC 10 · 2013 · #2
Easy mode Grade 5Alice needs 221 cups of sugar. Her measuring cup holds only 41 cup. How many scoops does she need to reach exactly 221 cups?
Pick an answer.
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.
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
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) $20$
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
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.
💡 Dividing the total by the size of one scoop tells you how many scoops it takes.
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.
💡 A whole cup holds four quarter-cups, so counting in quarters just means counting the pieces.
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 $\textbf{(B)}\ 10$.
💡 If ten quarter-cups make the total, then ten fills is the answer.
5.NF.B.7 Each fill adds $\frac{1}{4}$ cup, and the amounts pile up equally, so the number 4.NF.B.3 Count in $\frac{1}{4}$-cups. Each whole cup is four $\frac{1}{4}$-cups, so the $ 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.7Divide 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.3Understand 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.
⭐ 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.
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