1998 AMC 8 Problem 10

Attempt Problem 10 of the 1998 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1998 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

10.

Each of the letters W,W, X,X, Y,Y, and ZZ represents a different integer in the set {1,2,3,4},\{1,2,3,4\}, but not necessarily in that order. They satisfy WX−YZ=1\dfrac{W}{X}-\dfrac{Y}{Z}=1 What is the sum of WW and Y?Y?
33
44
55
66
77
Answer: E
Concepts:fractioncasework
Small hint:
List possible fractions using 1,1, 2,2, 3,3, and 4.4.
Big hint:
The larger fraction must exceed the smaller one by exactly 1.1.
Solution:
The only way to get a difference of 11 is
31−42=1.\frac{3}{1}-\frac{4}{2}=1.
Thus, W=3W=3 and Y=4,Y=4, so W+Y=7.W+Y=7.
Thus, the correct answer is E .
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