2023 AMC 10A Problem 2

Question

The weight of

\frac{1}{3}

of a large pizza together with

3\frac{1}{2}

cups of orange slices is the same as the weight of

\frac{3}{4}

of a large pizza together with

\frac{1}{2}

cup of orange slices. A cup of orange slices weighs

\frac{1}{4}

of a pound. What is the weight, in pounds, of a large pizza?

1\frac{4}{5}

2

2\frac{2}{5}

3

3\frac{3}{5}

Let

p

be the pizza’s weight and convert cups of orange slices to pounds using

\frac14

pound per cup

Set

\frac13 p + \frac72 \cdot \frac14 = \frac34 p + \frac12 \cdot \frac14

and solve for

p

Let

p

be the pizza’s weight. A cup of orange slices is

\frac14

pound, so the two sides balance as

\frac13 p + \frac72 \cdot \frac14 = \frac34 p + \frac12 \cdot \frac14,

that is

\frac13 p + \frac78 = \frac34 p + \frac18.

Collect the pizza terms:

\frac68 = \left(\frac34 - \frac13\right)p = \frac{5}{12}p.

So

p = \frac34 \cdot \frac{12}{5} = \frac95 = 1\frac45.

Therefore, the answer is A.

Problem 2 in other years