AMC 10 · 2015 · #4

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking

ratio-proportionfraction-arithmetic convert-to-algebra ↑ Prerequisites: ratio-proportion

📏 Short solution 💡 2 insights

Problem

Pablo has three times as many eggs as Sofia, and Sofia has twice as many as Mia. Pablo hands out some of his eggs so that all three end up with the same number. Find what fraction of his own eggs Pablo gives to Sofia.

Pick an answer.

(A)

$\frac{1}{12}$

(B)

$\frac{1}{6}$

(C)

$\frac{1}{4}$

(D)

$\frac{1}{3}$

(E)

$\frac{1}{2}$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The two "times as many" statements chain together, so Tool #4 (Introduce a Variable) lets one letter for Mia's eggs carry everyone's count: Sofia and Pablo become simple multiples. Tool #6 (Guess and Check) is the shortcut twin here — picking the smallest whole number for Mia keeps every count a whole number and avoids fractions until the very last step. Once the equal share is known, the question "what fraction of his eggs" is a single part-to-whole ratio, and Tool #3 (Eliminate Possibilities) confirms the answer is one of the small fractions offered.

1STEP 1

Chain the counts from Mia up

Let Mia = m; then Sofia = 2m and Pablo = 3× 2m = 6m.

Mia=m, Sofia=2m, Pablo=6m

2STEP 2

Find the equal share

The total stays m+2m+6m=9m, so each equal share is 3m eggs.

m+2m+6m=9m, 9m/3=3m

3STEP 3

Give Sofia just enough, then take the fraction

Sofia needs 3m-2m=m more from Pablo's 6m: fraction = 1/6 — choice (B).

3m-2m=m, m/6m=1/6 → (B)

Answer

1/6

Pick m=1: Mia 1, Sofia 2, Pablo 6, total 9, so each should have 3. Pablo gives Sofia 1 egg (to reach 3) and Mia 2 eggs, handing out 3 of his 6 in all and keeping 3 — everyone lands on 3, which checks. The one egg to Sofia out of six is 1/6, matching (B). Note 1/2 is the fraction of all his eggs he gives away total, not the share to Sofia, so (E) is the trap.

💡Key takeaway

Name the smallest pile with a letter, build everyone else from it, and the messy ratios turn into easy whole numbers.

Chain the counts from Mia up
Find the equal share
Give Sofia just enough, then take the fraction

Chain the counts from Mia up

Find the equal share

Give Sofia just enough, then take the fraction

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