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2023 AMC 10A Problem 2

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2023 AMC 10A Problem 2AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

The weight of 13\frac{1}{3} of a large pizza together with 3123 \frac{1}{2} cups of orange slices is the same weight of 34\frac{3}{4} of a large pizza together with 12\frac{1}{2} cups of orange slices. A cup of orange slices weigh 14\frac{1}{4} of a pound. What is the weight, in pounds, of a large pizza?

  1. A)

    1451\frac{4}{5}

  2. B)

    22

  3. C)

    2252\frac{2}{5}

  4. D)

    33

  5. E)

    3353\frac{3}{5}

Answer

A

Key insight

Move the pizza terms to one side and the orange terms to the other: 5/12 of a pizza weighs the same as 3 cups, which is 3/4 pound.

Solution

Let PP be the weight of a whole pizza in pounds. Each cup of orange slices weighs 14\frac14 pound, so the balance says
13P+72⋅14=34P+12⋅14.\frac13 P + \frac72 \cdot \frac14 = \frac34 P + \frac12 \cdot \frac14.

Rather than converting right away, first compare the two sides in cups: the left has 33 more cups of oranges than the right, and the right has 34−13=512\frac34 - \frac13 = \frac{5}{12} more pizza. So 512\frac{5}{12} of a pizza weighs the same as 33 cups of orange slices, which is 3⋅14=343 \cdot \frac14 = \frac34 pound.

Therefore
P=34⋅125=95=145.P = \frac34 \cdot \frac{12}{5} = \frac{9}{5} = 1\frac45.

The answer is (A) 145\boxed{\textbf{(A)}\ 1\frac{4}{5}}.

Why this works

A balance statement is a linear equation; subtracting the common parts from both sides isolates "some fraction of a pizza equals some number of cups." Converting units only at the very end keeps the arithmetic to a single multiplication. This "difference of the two sides" habit is the standard way to solve any "aa of X plus bb of Y equals cc of X plus dd of Y" problem.

The trap

Forgetting to convert cups to pounds at the end, or mishandling the mixed number 3 1/2.

Common mistakes

  • Forgetting to convert cups to pounds at the end, or mishandling the mixed number 3 1/2.
  • Subtracting the fractions of pizza incorrectly (34−13=512\frac34 - \frac13 = \frac{5}{12}, not 24\frac{2}{4} or 12\frac{1}{2}).

Techniques

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