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

Two different baskets of the same two goods have totals in a known ratio. Find the price ratio.

Pick an answer.

(A)

$\frac{3}{2}$

(B)

$\frac{5}{3}$

(C)

$\frac{7}{4}$

(D)

(E)

$\frac{13}{4}$

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

Two letters price both baskets.

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

2STEP 2

Turn "twice as much" into an equation

The comparison becomes one equation.

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

3STEP 3

Collect like terms and read off the ratio

Collecting terms gives 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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