AMC 10 · 2020 · #3

Grade 6 rate-ratio

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

ratio-proportionfraction-arithmetic identify-subproblemsguess-and-check ↑ Prerequisites: ratio-proportion

📏 Short solution 💡 2 insights

Problem

Three ratios link four quantities: w:x = 4:3, y:z = 3:2, and z:x = 1:6. Find w:y.

Pick an answer.

(A)

4:3

(B)

3:2

(C)

8:3

(D)

4:1

(E)

16:3

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

Tool #6 (Guess and Check) via WLOG: because ratios are scale-free, we are allowed to pick one variable to be a convenient number that satisfies the cleanest denominators. Letting x = 6 makes both w:x = 4:3 and z:x = 1:6 produce whole numbers. Tool #7 (Subproblems): split into three little jumps — x → w, x → z, z → y, then form w:y. Tool #15 (Reorganize): the ratios are given as w{:}x, y{:}z, z{:}x — re-order them along the chain w arrow x → z → y so the path from w to y is obvious.

1STEP 1

Pick a convenient value: let x = 6 so both z:x = 1:6 and w:x = 4:3 give whole numbers.

x = 6

2STEP 2

From w:x = 4:3 with x = 6, scale by 2 to get w = 8.

w:x = 4:3 = 8:6 → w = 8

3STEP 3

From z:x = 1:6 with x = 6, read off z = 1.

z:x = 1:6 → z = 1

4STEP 4

From y:z = 3:2 with z = 1, halve to get y = $\frac{3}{2}$ .

y:z = 3:2 → y =

\frac{3}{2}

z =

\frac{3}{2}

5STEP 5

Form w:y = 8 : $\frac{3}{2}$ , then double both parts to clear the fraction: 16 : 3.

w:y = 8 :

\frac{3}{2}

= 16 : 3 → (E)

Answer

16:3

Try a different starting value. Let x = 12: then w = 16, z = 2, y = 3. Check w:y = 16:3 — same ratio, confirming (E). Sanity: w is much bigger than the others while y is small, so w:y ought to be large; 16:3 matches that feel.

💡Key takeaway

This AMC 10 problem only needs Grade 6 “ratios scale up and down the same way” you already know — set x = 6 to get w = 8, z = 1, y = $\frac{3}{2}$ , then w:y = 16:3.

More like this