The point $(\frac{1}{2},\frac{-13}{4})$ divides the line segment joining the points $(3,-5)$ and $(-7,2)$ in the ratio

$[\frac{\left(m\right)(-7)+n\left(3\right)}{m+n},\frac{\left(m\right)\left(2\right)+n(-5)}{m+n}]=[\frac{1}{2},\frac{-13}{4}]$

$\frac{-7m+3n}{m+n}=\frac{1}{2}$

$-14m+6n=m+n$

$-15m=-5n$

$\frac{m}{n}=\frac{1}{3}$

$m:n=1:3\text{\hspace{0.17em}internally}$

(OR) METHOD 2:

Ration = $x_1-x:x-x_2$

$=3-\frac{1}{2}:\frac{1}{2}+7$

$=\frac{5}{2}:\frac{15}{2}=5:15=1:3$ internally

(OR) METHOD 3:

$3-\frac{1}{2}:\frac{1}{2}+7$

$5:15$

$1:3$