2023 AMC 10A Problem 2

Think of the sentence as a balance scale with the same weight on each side.

1. Left side: $\frac13$ of a pizza and $3\frac12$ cups of orange slices.
Right side: $\frac34$ of a pizza and $\frac12$ cup of orange slices.
2. Take $\frac12$ cup of orange slices off both sides. The scale stays balanced. Now the left has $3$ cups and the right has none.
3. Take $\frac13$ of a pizza off both sides. The left has no pizza left, and the right has
$$\frac34 - \frac13 = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \text{ of a pizza}.$$
4. So $3$ cups of orange slices weigh the same as $\frac{5}{12}$ of a pizza.
5. Now change cups into pounds. One cup weighs $\frac14$ pound, so $3$ cups weigh $\frac34$ pound. That means $\frac{5}{12}$ of a pizza weighs $\frac34$ pound.
6. One twelfth of the pizza weighs $\frac34 \div 5 = \frac{3}{20}$ pound, so the whole pizza weighs
$$12 \times \frac{3}{20} = \frac{36}{20} = \frac95 = 1\frac45 \text{ pounds}.$$

The answer is $\boxed{\textbf{(A)}\ 1\frac{4}{5}}$.

A "these weigh the same" sentence is an equation, and you may remove the same thing from both sides. Removing the shared orange slices and the shared pizza leaves one clean fact: a certain piece of pizza equals a certain number of cups. Waiting until the end to turn cups into pounds keeps the fractions simple.

Write it as an equation with $P$ for the pizza's weight in pounds:
$$\frac13 P + \frac72 \cdot \frac14 = \frac34 P + \frac12 \cdot \frac14.$$
Multiply every term by $24$ to clear the fractions: $8P + 21 = 18P + 3$. Then $10P = 18$, so $P = 1.8 = 1\frac45$.