2023 AMC 10B Problem 1

P1-10~1 minPrint

Ratios, Percents & Averages Fractions & Decimals Set up the equation/formula and compute; no special trick needed

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 $\frac{1}{3}$ 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?

Show solution without answering

Key insight

Find the level every glass must end at first: the total juice split four ways. Then each full glass gives up the difference between full and that level.

Solution

The question asks how much to move, but it is easier to first work out where every glass should end up. Measure all amounts in "glasses".

1. Add up the juice. Three full glasses and one glass that is $\frac{1}{3}$ full:
$1 + 1 + 1 + \frac{1}{3} = \frac{10}{3} \text{ glasses}.$
2. Share it equally. Four glasses get the same amount, so each gets
$\frac{10}{3} \div 4 = \frac{10}{12} = \frac{5}{6} \text{ of a glass}.$
3. Find how much each full glass gives away. A full glass has to drop from $1$ to $\frac{5}{6}$ , so it gives away
$1 - \frac{5}{6} = \frac{1}{6}.$
4. Check the fourth glass. It gets $\frac{1}{6}$ from each of three glasses, which is $\frac{3}{6} = \frac{1}{2}$ . It started at $\frac{1}{3}$ , and $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ . That matches the other three glasses.

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

Why this works

"Make all the amounts equal" means every container ends at the average: the total divided by the number of containers. Once you know that final level, the amount to move is just the gap between where a glass starts and where it has to end. Find the ending level first, then the transfer.

Alternative approach

Call the amount poured out of each full glass $x$ .

Each full glass ends with $1 - x$ .
The fourth glass gets $3x$ in total, so it ends with $\frac{1}{3} + 3x$ .

These must be equal: $1 - x = \frac{1}{3} + 3x$ . Move the terms around to get $4x = \frac{2}{3}$ , so $x = \frac{2}{12} = \frac{1}{6}$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Check: the fourth glass receives $3 \cdot \frac{1}{6} = \frac{1}{2}$ , and $\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$ , matching the others.

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Number Theory, Chapter 11: Decimals and Fractions
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 6: Decimals (§6.3, §6.4)