AMC 10 · 2014 · #4

Grade 8 algebrarate-ratio

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

systems-of-equationsratio-proportion convert-to-algebra ↑ Prerequisites: systems-of-equations

📏 Short solution 💡 2 insights

Problem

Susie buys 4 muffins and 3 bananas. Calvin buys 2 muffins and 16 bananas and pays exactly twice as much as Susie. Every muffin costs the same and every banana costs the same. A muffin is how many times as expensive as a banana?

Pick an answer.

(A)

$\frac{3}{2}$

(B)

$\frac{5}{3}$

(C)

$\frac{7}{4}$

(D)

(E)

$\frac{13}{4}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

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 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 5/3 fits the arithmetic.

1STEP 1

Name the prices and write each total

Let m be a muffin's price and b a banana's price. Susie pays 4m+3b; Calvin pays 2m+16b.

Susie=4m+3b, Calvin=2m+16b

2STEP 2

Turn "twice as much" into an equation

Calvin's total is 2 times Susie's, so 2m+16b=2(4m+3b); distributing the 2 makes the right side 8m+6b.

2m+16b=2(4m+3b)=8m+6b

3STEP 3

Collect like terms and read off the ratio

Subtract 2m and 6b from both sides: 10b=6m, so dividing by 6b gives m/b = 10/6 = 5/3 — choice (B).

2m+16b=8m+6b → 10b=6m → m/b=10/6=5/3 → (B)

Answer

5/3

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 5/3≈1.67 is comfortably above 1, and it is a tidy fraction, both signs it is right. Choice (D) 2 or (E) 13/4 would make muffins too expensive to fit 10b=6m.

💡Key takeaway

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 the prices and write each total
Turn "twice as much" into an equation
Collect like terms and read off the ratio

Name the prices and write each total

Turn "twice as much" into an equation

Collect like terms and read off the ratio

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