AMC 10 · 2006 · #3

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionlinear-equations-one-var convert-to-algebra ↑ Prerequisites: ratio-proportion

📏 Short solution 💡 1 insight

Problem

Two ages are in the ratio three to five, and the older is 30. Find the younger age.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

A ratio 3:5 is really a set of equal-sized shares: Mary gets 3 shares and Alice gets 5. Tool #4 (Introduce a Variable) names the size of one share x, so Mary =3x and Alice =5x. Tool #13 (Convert to Algebra) turns the fact 'Alice is 30' into the equation 5x=30, which pins down x. Tool #3 (Eliminate Possibilities) gives a fast reality check: Mary is the smaller side of the ratio, so she must be younger than 30, which instantly kills (E) 50.

1STEP 1

Split the ages into equal shares

The ratio splits both ages into equal shares.

Mary = 3x, Alice = 5x

2STEP 2

Use Alice's age to size one share

The known age makes one share 6.

5x = 30 → x = 30/5 = 6

3STEP 3

Build Mary's age and confirm

Rebuilding gives 18, and the ratio checks out, choice (B).

Mary = 3x = 3 × 6 = 18 → (B)

Answer

Mary is the smaller side of the ratio, so her age must be below Alice's 30 — this rules out (E) 50 at a glance and tells us the answer sits in the low range. Testing the result: 18:30 divides both by 6 to give 3:5, exactly the ratio in the problem, so 18 is consistent. The other small choices fail this test — for example 15:30 = 1:2 and 20:30 = 2:3, neither of which is 3:5.

💡Key takeaway

A ratio splits both amounts into equal shares — find the size of one share from the number you know, then multiply to get the one you want.

Split the ages into equal shares
Use Alice's age to size one share
Build Mary's age and confirm

Split the ages into equal shares

Use Alice's age to size one share

Build Mary's age and confirm

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