Check the picture below.
let's bear in mind that the segment MN is simply the sum of MQ + QN, and since M and N are midpoints, they cut that respective section into two equal halves.
[tex]\bf \begin{cases} PQ=5QR\\ MN = \frac{PQ}{2}+\frac{QR}{2} \end{cases}\qquad \qquad \cfrac{PQ}{MN}\implies \cfrac{5QR}{\frac{PQ}{2}+\frac{QR}{2}}\implies \cfrac{5QR}{\frac{PQ+QR}{2}} \\\\\\ \cfrac{\frac{5QR}{1}}{\frac{PQ+QR}{2}}\implies \cfrac{5QR}{1}\cdot \cfrac{2}{PQ+QR}\implies \cfrac{10QR}{\underline{PQ}+QR}\implies \cfrac{10QR}{\underline{5QR}+QR}[/tex]
[tex]\bf \cfrac{~~\begin{matrix} 10QR \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{~~\begin{matrix} 6QR \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies \cfrac{5}{3}[/tex]
