Question

The answer is 5/3, but I don’t know the solution.

Answer 1

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.

`\bf \begin{cases} PQ=5QR\\ MN = (PQ)/(2)+(QR)/(2) \end{cases}\qquad \qquad \cfrac{PQ}{MN}\implies \cfrac{5QR}{(PQ)/(2)+(QR)/(2)}\implies \cfrac{5QR}{(PQ+QR)/(2)} \\\\\\ \cfrac{(5QR)/(1)}{(PQ+QR)/(2)}\implies \cfrac{5QR}{1}\cdot \cfrac{2}{PQ+QR}\implies \cfrac{10QR}{\underline{PQ}+QR}\implies \cfrac{10QR}{\underline{5QR}+QR}`

`\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}`