7. Lucy is asked to choose $p, q, r$ and $s$ to be the numbers $1,2,3$ and 4 , in some order, so as to make the value of $\frac{p}{q}+\frac{r}{s}$ as small as possible.
What is the smallest value Lucy can achieve in this way?
A $\frac{7}{12}$
B $\frac{2}{3}$
C $\frac{3}{4}$
D $\frac{5}{6}$
E $\frac{11}{12}$
Solution
D
To make the value of $\frac{p}{q}+\frac{r}{s}$ as small as possible, $q$ and $s$ need to be as large as possible, and so have the values 3 and 4 .
Therefore the expression with the smallest value that Lucy can achieve is either $\frac{1}{3}+\frac{2}{4}$ or $\frac{2}{3}+\frac{1}{4}$.
Now,
$$
\begin{aligned}
\frac{1}{3}+\frac{2}{4} & =\frac{1}{3}+\frac{1}{4}+\frac{1}{4} \\
& <\frac{1}{3}+\frac{1}{3}+\frac{1}{4} \\
& =\frac{2}{3}+\frac{1}{4} .
\end{aligned}
$$

It follows that the smallest value that Lucy can achieve is
$$
\frac{1}{3}+\frac{2}{4}=\frac{4+6}{12}=\frac{10}{12}=\frac{5}{6} \text {. }
$$