Factorise completely $5pr - 2ps - 5qr + 2qs$. Given that $p \neq q$, find the value of $\frac{r}{s}$ when $5pr - 2ps - 5qr + 2qs = 0$.

1. Group terms and factorise each pair

$5pr-2ps-5qr+2qs$

$=p(5r-2s)-q(5r-2s)$

2. Extract the common binomial factor

$p(5r-2s)-q(5r-2s)$

$=(p-q)(5r-2s)$

1. Substitute the factorised form from part (a) into the equation

From part (a),
$5pr-2ps-5qr+2qs=(p-q)(5r-2s).$
So
$(p-q)(5r-2s)=0.$

2. Use $p\neq q$ and solve for $\frac{r}{s}$

Since $p\neq q$, $p-q\neq 0$.

Hence
$5r-2s=0.$
$5r=2s$
$\frac{r}{s}=\frac{2}{5}.$