AMC 10 Step by Step
Log inSign up
Printable

2023 AMC 10B Problem 1

Problem onlyBack to the problem
2023 AMC 10B Problem 1AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only 13\frac{1}{3} full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

  1. A)

    112\frac{1}{12}

  2. B)

    14\frac{1}{4}

  3. C)

    16\frac{1}{6}

  4. D)

    18\frac{1}{8}

  5. E)

    29\frac{2}{9}

Answer

C

Key insight

Total juice is 3 + 1/3 = 10/3 glasses; shared equally each glass holds 5/6, so each full glass gives up 1/6.

Solution

Measure everything in glasses. The total amount of juice is
3+13=103.3 + \frac{1}{3} = \frac{10}{3}.
If the four glasses are to hold equal amounts, each must hold
103÷4=1012=56.\frac{10}{3} \div 4 = \frac{10}{12} = \frac{5}{6}.
A full glass must therefore be lowered from 11 to 56\frac{5}{6}, so Mrs. Jones pours 1−56=161 - \frac{5}{6} = \frac{1}{6} of a glass out of each of the first three.

Check: the fourth glass receives 3⋅16=123 \cdot \frac{1}{6} = \frac{1}{2}, and 13+12=56\frac{1}{3} + \frac{1}{2} = \frac{5}{6}, matching the others.

The answer is (C) 16\boxed{\textbf{(C)}\ \frac{1}{6}}.

Why this works

"Make all amounts equal" is an averaging problem: the common amount is the total divided by the number of containers, and the amount moved is the gap between a full glass and that average. Compute the final level first, then the transfer; that order avoids guessing.

Alternative approach

Let xx be the amount poured from each full glass. The full glasses end at 1−x1 - x and the fourth ends at 13+3x\frac{1}{3} + 3x. Setting them equal, 1−x=13+3x1 - x = \frac{1}{3} + 3x, so 4x=234x = \frac{2}{3} and x=16x = \frac{1}{6}.

The trap

Splitting the missing 2/3 of the fourth glass three ways (2/9) without realizing the target level is 5/6, not a full glass.

Common mistakes

  • Splitting the missing 2/3 of the fourth glass three ways (2/9) without realizing the target level is 5/6, not a full glass.
  • Dividing the total 103\frac{10}{3} by 33 instead of 44, forgetting the fourth glass counts as one of the equal shares.

Techniques

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