AMC 8 · 2011 · #1

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

multi-digit-arithmeticunit-conversionmental-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?

Pick an answer.

(A)

$\textdollar 1.50$

(B)

$\textdollar 2.00$

(C)

$\textdollar 2.50$

(D)

$\textdollar 3.00$

(E)

$\textdollar 3.50$

AMC 8 2011 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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Margie buys $3$ apples that cost $50$ cents each and pays with a $\$5$ bill. How much change does she get back?

Givens: Number of apples bought $= 3$; Price per apple $= 50$ cents; Amount paid $= \$5$ bill; Answer choices: (A) $\$1.50$, (B) $\$2.00$, (C) $\$2.50$, (D) $\$3.00$, (E) $\$3.50$

Unknowns: The amount of change Margie receives, in dollars

Understand

Restated: Margie buys $3$ apples that cost $50$ cents each and pays with a $\$5$ bill. How much change does she get back?

Givens: Number of apples bought $= 3$; Price per apple $= 50$ cents; Amount paid $= \$5$ bill; Answer choices: (A) $\$1.50$, (B) $\$2.00$, (C) $\$2.50$, (D) $\$3.00$, (E) $\$3.50$

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

The only catch in this problem is mixed units — cents for the apple price, dollars for the bill and the answer choices. Tool #8 (Analyze the Units) makes sure we convert $50$ cents into $\$0.50$ (or $150$ cents into $\$1.50$) before subtracting. Tool #7 (Identify Subproblems) breaks the work into two clean pieces: first find the total cost of the apples, then subtract from $\$5$ to get the change.

Execute — Answer: E

#7 Identify Subproblems 3.OA.A.3 Step 1

Find the total cost of the apples.
Three apples at $50$ cents each is a multiplication: number of apples times cost per apple.

$$3 \times 50 \text{ cents} = 150 \text{ cents}$$

💡 Multiplying "how many" by "price each" to get a total is the Grade 3 multiplication word-problem move.

#8 Analyze the Units 4.MD.A.1 Step 2

Convert the total cost from cents to dollars so it matches the units of the $\$5$ bill. Since $100$ cents $= \$1$, $150$ cents $= \$1.50$.

$150 \text{ cents} \times \dfrac{\$1}{100 \text{ cents}} = \$1.50$

💡 Switching between cents and dollars within the same money system is the Grade 4 measurement-conversion skill.

#7 Identify Subproblems 4.MD.A.2 Step 3

Subtract the total cost from the amount paid to find the change.

$\$5.00 - \$1.50 = \$3.50 \;\Rightarrow\; \textbf{(E)}$

💡 Change $=$ paid $-$ cost is the classic money word-problem subtraction at Grade 4.

[1] #7 3.OA.A.3 Find the total cost of the apples. Three apples at $50$ cents each is a multipli

[2] #8 4.MD.A.1 Convert the total cost from cents to dollars so it matches the units of the $\$5

[3] #7 4.MD.A.2 Subtract the total cost from the amount paid to find the change.

Review

Reasonableness: Three apples at about $50$ cents each is around $\$1.50$ total, which is much less than $\$5$, so the change should be most of the bill — closer to $\$3.50$ than to $\$1.50$. The answer (E) $\$3.50$ fits that estimate exactly.

Alternative: Tool #6 (Guess and Check) on the choices: if the change were $\$3.50$, then the apples cost $\$5.00 - \$3.50 = \$1.50$, i.e. $50$ cents each — which matches the problem. The other choices give apple totals of $\$3.50$, $\$3.00$, $\$2.50$, $\$2.00$, none of which equal $3 \times \$0.50$.

CCSS standards used (min grade 4)

3.OA.A.3 Use multiplication and division within $100$ to solve word problems (Multiplying $3 \times 50 = 150$ cents to find the total cost of the apples.)
4.MD.A.1 Convert measurements from a larger unit to a smaller unit within the same measurement system (Converting $150$ cents into $\$1.50$ so the total cost and the payment use the same unit.)
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, masses, and money (Subtracting $\$5.00 - \$1.50 = \$3.50$ to find Margie's change in a money word problem.)

⭐ This AMC 8 problem only needs Grade 4 money sense — convert cents to dollars, then subtract — that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

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