AMC 10 · 2014 · #4
Grade 8 arithmeticSusie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
Pick an answer.
AMC 10 2014 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: Susie buys $4$ muffins and $3$ bananas. Calvin pays exactly twice as much for $2$ muffins and $16$ bananas. Using one price for every muffin and one price for every banana, find how many times the price of a muffin is compared to the price of a banana.
Givens: Susie's cost is for $4$ muffins and $3$ bananas; Calvin's cost is for $2$ muffins and $16$ bananas; Calvin spends twice as much as Susie; Answer choices: (A) $\dfrac{3}{2}$, (B) $\dfrac{5}{3}$, (C) $\dfrac{7}{4}$, (D) $2$, (E) $\dfrac{13}{4}$
Unknowns: The ratio of a muffin's price to a banana's price
Understand
Restated: Susie buys $4$ muffins and $3$ bananas. Calvin pays exactly twice as much for $2$ muffins and $16$ bananas. Using one price for every muffin and one price for every banana, find how many times the price of a muffin is compared to the price of a banana.
Givens: Susie's cost is for $4$ muffins and $3$ bananas; Calvin's cost is for $2$ muffins and $16$ bananas; Calvin spends twice as much as Susie; Answer choices: (A) $\dfrac{3}{2}$, (B) $\dfrac{5}{3}$, (C) $\dfrac{7}{4}$, (D) $2$, (E) $\dfrac{13}{4}$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The prices are unknown, so Tool #4 (Introduce a Variable) names a muffin $m$ and a banana $b$ and writes each shopper's total as an expression. Tool #13 (Convert to Algebra) turns the phrase "twice as much" into a single equation, which after collecting like terms gives the ratio $\dfrac{m}{b}$ directly — the two prices cancel, so no dollar amount is needed. Tool #3 (Eliminate Possibilities) is a fast sanity filter: muffins clearly cost more than bananas here, so the ratio must be more than $1$, and only a clean value like $\dfrac{5}{3}$ fits the arithmetic.
Execute — Answer: B
6.EE.A.2 Step 1 Name the prices and write each total
- Let $m$ be the price of one muffin and $b$ the price of one banana.
- Susie's purchase of $4$ muffins and $3$ bananas costs $4m+3b$.
- Calvin's purchase of $2$ muffins and $16$ bananas costs $2m+16b$.
💡 Giving each unknown price a letter lets you write the totals exactly instead of guessing dollar amounts.
6.EE.A.3 Step 2 Turn "twice as much" into an equation
- Calvin spends twice as much as Susie, so Calvin's total equals $2$ times Susie's total.
- Write $2m+16b=2(4m+3b)$, then use the distributive property to expand the right side: $2(4m+3b)=8m+6b$.
💡 "Twice as much" is just multiplication by $2$, and distributing that $2$ clears the parentheses so both sides are plain sums.
8.EE.C.7 Step 3 Collect like terms and read off the ratio
- Gather the muffin terms on one side and the banana terms on the other: subtract $2m$ and $6b$ from both sides to get $10b=6m$.
- Divide both sides by $6b$ to isolate the ratio: $\dfrac{m}{b}=\dfrac{10}{6}=\dfrac{5}{3}$.
- So a muffin costs $\dfrac{5}{3}$ times as much as a banana, which is choice (B).
- This value is greater than $1$, matching the earlier check that muffins are the pricier item.
💡 Once every muffin term is on one side and every banana term on the other, dividing turns the equation straight into the price ratio.
6.EE.A.2 Let $m$ be the price of one muffin and $b$ the price of one banana. Susie's purc 6.EE.A.3 Calvin spends twice as much as Susie, so Calvin's total equals $2$ times Susie's 8.EE.C.7 Gather the muffin terms on one side and the banana terms on the other: subtract Review
Reasonableness: The answer should be more than $1$ because Calvin swaps a muffin for lots of bananas ($2$ muffins vs Susie's $4$) yet still pays double — that only works if muffins are worth clearly more than bananas. The ratio $\dfrac{5}{3}\approx1.67$ is comfortably above $1$, and it is a tidy fraction, both signs it is right. Choice (D) $2$ or (E) $\dfrac{13}{4}$ would make muffins too expensive to fit $10b=6m$.
Alternative: Pick a convenient banana price and test. Let $b=3$; then $m=\dfrac{5}{3}\cdot3=5$. Susie pays $4(5)+3(3)=20+9=29$, and Calvin pays $2(5)+16(3)=10+48=58$. Since $58=2\times29$, Calvin does spend exactly twice as much, confirming the ratio $\dfrac{m}{b}=\dfrac{5}{3}$ and choice (B).
CCSS standards used (min grade 8)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Letting $m$ and $b$ stand for the prices and writing the totals $4m+3b$ and $2m+16b$.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Using the distributive property to expand $2(4m+3b)$ into $8m+6b$.)8.EE.C.7Solve linear equations in one variable (Collecting like terms in $2m+16b=8m+6b$ to get $10b=6m$ and solving for the ratio $\dfrac{m}{b}=\dfrac{5}{3}$.)
⭐ Name each unknown price with a letter, turn "twice as much" into an equation, then gather like terms — the prices cancel and the ratio falls right out.
⭐ Name each unknown price with a letter, turn "twice as much" into an equation, then gather like terms — the prices cancel and the ratio falls right out.
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