Competition · AMC preparation · step 4 of 4

AMC 10 · 2023B · #1

Grade 5 rate-ratio

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

fraction-arithmeticmean-median-mode-rangeratio-proportion easier-related-problemwork-backwardsidentify-subproblems ↑ Prerequisites: fraction-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

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Problem

Three glasses are full and a fourth holds only

\frac{1}{3}

of a glass. Find the fraction x that must be poured from each full glass into the fourth so that all four glasses end up with the same amount.

Pick an answer.

(A)

$\frac{1}{12}$

(B)

$\frac{1}{4}$

(C)

$\frac{1}{6}$

(D)

$\frac{1}{8}$

(E)

$\frac{2}{9}$

AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The easier problem hiding inside is just averaging: if all the juice were pooled and shared equally among 4 identical glasses, how much would each glass get? Tool #9 reframes the whole pouring scene as "find the average level". Once we know the target level, Tool #11 (Work Backwards) finishes: each full glass must drop from 1 to that target, and the amount poured out is the difference. A quick Tool #1 sketch of four bar-glasses makes the conservation visible without algebra.

1STEP 1

Add up all the juice

Pool all the juice — pouring only moves it, so the total is conserved at $\frac{10}{3}$ glasses.

Total = 1 + 1 + 1 + 1/3 = 10/3 glasses

2STEP 2

Share it evenly among four

Share that total equally among the four glasses; each should end at $\frac{5}{6}$ of a glass.

Target per glass = 10/3/4 = 10/3 · 1/4 = 10/12 = 5/6

3STEP 3

Find how much each glass gives

Work backwards: each full glass must drop from 1 to $\frac{5}{6}$ , so pour out the difference, $\frac{1}{6}$ of a glass.

x = 1 - 5/6 = 6/6 - 5/6 = 1/6

4STEP 4

Check the fourth glass

Check the fourth glass: three pours of $\frac{1}{6}$ add $\frac{1}{2}$ , lifting $\frac{1}{3}$ to $\frac{5}{6}$ — matching the other three. Answer (C).

1/3 + 3 · 1/6 = 2/6 + 3/6 = 5/6 → (C)

Answer

1/6

Sanity-check the magnitudes. The fourth glass is short by

\frac{5}{6}

\frac{1}{3}

\frac{1}{2}

of a glass; splitting that shortage equally among the three donor glasses gives

\frac{1}{2}

÷ 3 =

\frac{1}{6}

per donor — matching the answer. Also,

\frac{1}{6}

is small enough that each donor still has plenty left (

\frac{5}{6}

), which fits the physical picture.

💡Key takeaway

This AMC 10 problem only needs Grade 5 fraction-sharing you already know — pool all the juice, divide by 4, and pour out the leftover above that target.

Add up all the juice
Share it evenly among four
Find how much each glass gives
Check the fourth glass

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Add up all the juice

Share it evenly among four

Find how much each glass gives

Check the fourth glass

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