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2020 AMC 10B Problem 3

2020 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 2 min · position P1-10

The ratio of $w$ to $x$ is $4:3$ , the ratio of $y$ to $z$ is $3:2$ , and the ratio of $z$ to $x$ is $1:6$ . What is the ratio of $w$ to $y?$

A)
$4:3$
B)
$3:2$
C)
$8:3$
D)
$4:1$
E)
$16:3$

Answer

Key insight

Pick a concrete value for x that fits every ratio (x = 6 makes z = 1), then read off w and y.

Solution

The variable $x$ appears in two ratios, so choose a value for it that keeps everything integral. Take $x = 6$ .

From $w : x = 4 : 3$ we get $w = \tfrac{4}{3} \cdot 6 = 8$ .

From $z : x = 1 : 6$ we get $z = 1$ .

From $y : z = 3 : 2$ we get $y = \tfrac{3}{2} \cdot 1 = \tfrac{3}{2}$ .

Therefore
$\frac{w}{y} = \frac{8}{3/2} = \frac{16}{3},$
so $w : y = 16 : 3$ .

The answer is $\boxed{\textbf{(E)}\ 16:3}$ .

Why this works

Ratios only fix relative sizes, so you are free to assign one convenient number and let the rest follow. Choosing the value that is linked to the most ratios (here $x$ ) minimizes fractions. Equivalently, $\frac{w}{y} = \frac{w}{x}\cdot\frac{x}{z}\cdot\frac{z}{y} = \frac43 \cdot 6 \cdot \frac23 = \frac{16}{3}$ : chain the fractions so intermediate variables cancel.

The trap

Inverting one of the ratios, especially reading z:x = 1:6 as x:z = 1:6, which flips the answer.

Common mistakes

Inverting one of the ratios, especially reading z:x = 1:6 as x:z = 1:6, which flips the answer.
Chaining the fractions in the wrong direction (multiplying $\frac{w}{x}$ by $\frac{z}{x}$ instead of $\frac{x}{z}$ ) and getting $\frac{4}{3}\cdot\frac{1}{6}\cdot\frac{2}{3}$ .

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer