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2019 AMC 10B Problem 1

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2019 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

Alicia had two containers. The first was 56\tfrac{5}{6} full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was 34\tfrac{3}{4} full of water. What is the ratio of the volume of the first container to the volume of the second container?

  1. A)

    58\frac{5}{8}

  2. B)

    45\frac{4}{5}

  3. C)

    78\frac{7}{8}

  4. D)

    910\frac{9}{10}

  5. E)

    1112\frac{11}{12}

Answer

D

Key insight

The same amount of water is 5/6 of the first container and 3/4 of the second, so the volume ratio is (3/4) divided by (5/6).

Solution

Let the containers have volumes V1V_1 and V2V_2. The water is the same before and after pouring, so
56V1=34V2.\frac{5}{6}V_1 = \frac{3}{4}V_2.
Solve for the requested ratio:
V1V2=3/45/6=34⋅65=1820=910.\frac{V_1}{V_2} = \frac{3/4}{5/6} = \frac{3}{4}\cdot\frac{6}{5} = \frac{18}{20} = \frac{9}{10}.

The answer is (D) 910\boxed{\textbf{(D)}\ \frac{9}{10}}.

Why this works

One physical quantity (the water) is described as two different fractions of two different wholes. Setting the two descriptions equal gives one equation in the two volumes, and a ratio is all that equation can determine, which is exactly what is asked. Whenever a problem says "the same amount is pp of this and qq of that," the ratio of the wholes is q/pq/p.

Alternative approach

Pick a convenient number: suppose the water is 1515 liters (a multiple of both 55 and 33). Then V1=15⋅65=18V_1 = 15 \cdot \frac{6}{5} = 18 and V2=15⋅43=20V_2 = 15 \cdot \frac{4}{3} = 20, so the ratio is 18:20=9:1018 : 20 = 9 : 10.

The trap

Dividing the fractions in the wrong order and getting 10/9, or subtracting them instead of dividing.

Common mistakes

  • Dividing the fractions in the wrong order and getting 10/9, or subtracting them instead of dividing.
  • Reporting the ratio of the second container to the first because the second was mentioned last.

Techniques

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