2015 AMC 10A Problem 4

P1-10~2 minPrint

Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed Test small/specific values or special cases to find or verify the answer

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

Show solution without answering

Key insight

Only the ratio matters, so give Mia 1 egg, Sofia 2 and Pablo 6. Everyone should end with 3, so Sofia gets 1 of Pablo's 6 eggs.

Solution

The question asks for a fraction, so the actual number of eggs does not matter. Pick the smallest numbers that fit the story.

Give Mia $1$ egg. Sofia has twice as many, so $2$ eggs. Pablo has three times Sofia's, so $3 \times 2 = 6$ eggs.
All together they have $1 + 2 + 6 = 9$ eggs. For everyone to have the same number, each person must end with $9 \div 3 = 3$ eggs.
Sofia goes from $2$ to $3$ , so Pablo gives her $1$ egg. (Mia goes from $1$ to $3$ , so she gets $2$ . Pablo keeps $6 - 1 - 2 = 3$ . Good.)
The question asks what fraction of Pablo's eggs goes to Sofia: $1$ egg out of his $6$ , which is $\frac{1}{6}$ .

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

Why this works

When only ratios are given and only a fraction is asked, choose easy numbers that fit the ratios. That turns the story into small, concrete arithmetic. Before answering, reread the question: here it wants Sofia's share only, measured against Pablo's starting pile.

Concise solution

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

Only ratios matter, so choose convenient numbers. Let Mia have $1$ egg. Then Sofia has $2$ and Pablo has $3 \cdot 2 = 6$ .

The total is $1 + 2 + 6 = 9$ eggs, so after the redistribution each person holds $3$ . Sofia must go from $2$ to $3$ , receiving $1$ egg from Pablo. (Mia receives $2$ .)

Pablo gives Sofia $1$ of his $6$ eggs, which is the fraction $\frac{1}{6}$ .

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)