2020 AMC 10B Problems/Problem 3

Problem 3

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 $y$ ?

$\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\ 8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3$

Solution 1

WLOG, let $w=4$ and $x=3$ .

Since the ratio of $z$ to $x$ is $1:6$ , we can substitute in the value of $x$ to get $\frac{z}{3}=\frac{1}{6} \implies z=\frac{1}{2}$ .

The ratio of $y$ to $z$ is $3:2$ , so $\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}$ .

The ratio of $w$ to $y$ is then $\frac{4}{\frac{3}{4}}=\frac{16}{3}$ so our answer is $\boxed{\textbf{(E)}\ 16:3}$ ~quacker88

Solution 2

We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.

$z:x=1:6=2:12$ , and since $y:z=3:2$ , we can link them together to get $y:z:x=3:2:12$ .

Finally, since $x:w=3:4=12:16$ , we can link this again to get: $y:z:x:w=3:2:12:16$ , so $w:y = \boxed{\textbf{(E)}\ 16:3}$ ~quacker88

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2020 AMC 10B Problems/Problem 3

Problem 3

Solution 1

Solution 2