AMC 10 · 2019 · #1

Grade 5 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

ratio-proportionfraction-multiplicationfraction-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Short solution 💡 2 insights

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Problem

Alicia has two containers. The first starts

\frac{5}{6}

full of water; the second is empty. She pours all the water from the first into the second, which then becomes

\frac{3}{4}

full. Find the ratio of the first container's volume to the second container's volume.

Pick an answer.

(A)

$\frac{5}{8}$

(B)

$\frac{4}{5}$

(C)

$\frac{7}{8}$

(D)

$\frac{9}{10}$

(E)

$\frac{11}{12}$

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Tool #9 (Easier Problem): pick a friendly number for the second container's volume (say V₂ = 12, the LCM of 6 and 4) so both fractions give whole-number amounts. Then read V₁ off directly. Tool #8 (Units): water amount poured equals water amount received, so V₁ · $\frac{5}{6}$ = V₂ · $\frac{3}{4}$ — solving for $\frac{V₁}{V₂}$ falls right out. Tool #3 (Eliminate): the first container holds less water at $\frac{5}{6}$ full than the second at $\frac{3}{4}$ full only if V₁ ＜ V₂, so the ratio is less than 1 — all five choices satisfy this, but the clean fraction match picks (D).

1STEP 1

Let V₂ = 12 (the LCM of 6 and 4), so the poured water is $\frac{3}{4}$ · 12 = 9 units.

V₂ = 12 → water poured in =

\frac{3}{4}

· 12 = 9

2STEP 2

All 9 units came from the first container, which was $\frac{5}{6}$ full, so $\frac{5}{6}$ · V₁ = 9.

\frac{5}{6}

· V₁ = 9

3STEP 3

Undo the $\frac{5}{6}$ by multiplying: V₁ = 9 · $\frac{6}{5}$ = $\frac{54}{5}$ = 10.8 units.

V₁ = 9 ·

\frac{6}{5}

\frac{54}{5}

4STEP 4

Form the ratio V₁ : V₂ = $\frac{54}{5}$ : 12 = 54 : 60 = 9 : 10, so the ratio is $\frac{9}{10}$ (D).

\frac{V₁}{V₂}

\frac{\frac{54}{5}}{12}

\frac{54}{60}

\frac{9}{10}

→ (D)

Answer

\frac{9}{10}

Sanity check: the first container at

\frac{5}{6}

full holds the same water as the second at

\frac{3}{4}

full. Since

\frac{5}{6}

＞

\frac{3}{4}

, the same amount of water fills more of the first container — so the first must be smaller. The ratio

\frac{9}{10}

＜ 1 confirms this. Plugging back: V₁ = 10.8,

\frac{5}{6}

· 10.8 = 9; V₂ = 12,

\frac{3}{4}

· 12 = 9. Both sides match.

💡Key takeaway

This AMC 10 problem only needs Grade 5 fraction multiplication and division you already know — pick the second jar to be 12 units, then the water is 9 units, the first jar is 10.8 units, and the ratio is $\frac{9}{10}$ .

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