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2014 AMC 10B Problem 4

2014 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Susie 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?

A)
$\frac{3}{2}$
B)
$\frac{5}{3}$
C)
$\frac{7}{4}$
D)
$2$
E)
$\frac{13}{4}$

Answer

Key insight

Calvin's bill equals twice Susie's, so 2m + 16b = 8m + 6b, which collapses to 10b = 6m.

Solution

Let a muffin cost $m$ and a banana cost $b$ . Susie pays $4m + 3b$ ; Calvin pays $2m + 16b$ , and that is twice Susie's amount:
$2m + 16b = 2(4m + 3b) = 8m + 6b.$

Collect terms: $16b - 6b = 8m - 2m$ , so $10b = 6m$ , giving
$\frac{m}{b} = \frac{10}{6} = \frac{5}{3}.$

The answer is $\boxed{\textbf{(B)}\ \frac{5}{3}}$ .

Why this works

Two unknown prices with one relation between the bills cannot be solved individually, but the question only asks for their ratio, and a single homogeneous linear equation determines exactly that. When a problem asks "how many times as expensive," expect to solve for $m/b$ , not for $m$ and $b$ .

Alternative approach

Set $b = 1$ (only the ratio matters). Then $2m + 16 = 8m + 6$ gives $6m = 10$ , so $m = \frac53$ .

The trap

Setting the two bills equal instead of making Calvin's twice Susie's, or inverting the final ratio to get b/m.

Common mistakes

Setting the two bills equal instead of making Calvin's twice Susie's, or inverting the final ratio to get b/m.
Doubling the wrong side, writing $2(2m + 16b) = 4m + 3b$ , which gives a negative price.

Techniques

Set up the equation/formula and compute; no special trick needed