2014 AMC 10B Problem 4

1. Calvin spends twice as much as Susie. Twice Susie's order is $2 \times (4 \text{ muffins} + 3 \text{ bananas}) = 8$ muffins and $6$ bananas.
2. So these two baskets cost exactly the same:
- $2$ muffins and $16$ bananas (Calvin's order),
- $8$ muffins and $6$ bananas (double Susie's order).
3. Take away what the baskets have in common, $2$ muffins and $6$ bananas, from each. What is left still costs the same: $10$ bananas cost the same as $6$ muffins.
4. So one muffin costs as much as $\frac{10}{6} = \frac53$ bananas.

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

We cannot find the actual prices, because there is only one piece of information. But the question only asks how many times as expensive a muffin is, and one balance between two baskets is enough for that. Comparing baskets and cancelling what they share is the same as solving the equation $2m + 16b = 2(4m + 3b)$.

Pretend a banana costs $\$1$ (only the comparison matters, so any price works). If a muffin costs $m$ dollars, then Calvin pays $2m + 16$ and Susie pays $4m + 3$. Calvin pays twice as much, so $2m + 16 = 8m + 6$. Then $10 = 6m$ and $m = \frac53$ dollars, which is $\frac53$ times a banana.

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