Problem:
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 ?
Answer Choices:
A. 4:3
B. 3:2
C. 8:3
D. 4:1
E. 16:3
Solution:
We have
yw​= xw​⋅zx​⋅yz​= 34​⋅16​⋅32​= 316​
from which w:y= (E)16:3​
OR
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=(E)16:3​.
OR
We have the equations xw​=34​,zy​=23​, and xz​=61​. Clearing denominators, we have 3w=4x,2y=3z, and 6z=x. Since we want yw​, we look to find y in terms of x since we know the relationship between x and w. We begin by multiplying both sides of 2y=3z by two, obtaining 4y=6z. We then substitute that into 6z=x to get 4y=x. Now, to be able to substitute this into out first equation, we need to have 4x on the RHS. Multiplying both sides by 4 , we have 16y=4x. Substituting this into our first equation, we have 3w=16y, or yw​=316​, so our answer is (E)16:3​.
The problems on this page are the property of the MAA's American Mathematics Competitions