# 1994 AJHSME Problems/Problem 20

## Problem

Let $W,X,Y$ and $Z$ be four different digits selected from the set

$\{ 1,2,3,4,5,6,7,8,9\}.$

If the sum $\dfrac{W}{X} + \dfrac{Y}{Z}$ is to be as small as possible, then $\dfrac{W}{X} + \dfrac{Y}{Z}$ must equal

$\text{(A)}\ \dfrac{2}{17} \qquad \text{(B)}\ \dfrac{3}{17} \qquad \text{(C)}\ \dfrac{17}{72} \qquad \text{(D)}\ \dfrac{25}{72} \qquad \text{(E)}\ \dfrac{13}{36}$

## Solution

$$\frac{W}{X} + \frac{Y}{Z} = \frac{WZ+XY}{XZ}$$

Small fractions have small numerators and large denominators. To maximize the denominator, let $X=8$ and $Z=9$.

$$\frac{9W+8Y}{72}$$

To minimize the numerator, let $W=1$ and $Y=2$.

$$\frac{9+16}{72} = \boxed{\text{(D)}\rightarrow \frac{25}{72}}$$

Problem

Mr. Langry had two variables, $A$ and $B$. What is $A+B$?

$\text{(A)}\$A+B$\qquad \text{(B)}\ \dfrac{3}{17} \qquad \text{(C)}\ \dfrac{17}{72} \qquad \text{(D)}\ \dfrac{25}{72} \qquad \text{(E)}\ \dfrac{13}{36}$

Solution

The answer is simple, it is $\boxed{\text{(A)}\$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)A+B (Error compiling LaTeX. ! Missing \$ inserted.)