The point P(5, 2) lies on the segment joining A(1, -2) and B(8, 5). Find AP : PB.

Let AP : PB = $m:n$. From $x:5=\frac{8m+n}{m+n}$, so $5m+5n=8m+n$ and $4n=3m$, giving $\frac{m}{n}=\frac{4}{3}$. Check with $y:\frac{4\cdot 5+3\cdot (-2)}{7}=\frac{14}{7}=2$. The ratio is 4 : 3. The answer 3 : 4 reverses $m$ and $n$; 4 : 7 is AP : AB, a part against the whole; 5 : 2 just copies P's own coordinates.