The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers.
Let $m$ be the first positive integer.
Then $\frac{m}{6}$ is the second positive integer.
This gives $m = 21.4$, but now I'm stuck?
Assuming $x\geq y$ We have it that $$ 5(x+y)=7(x-y) $$
and that $x+y\leq25$
Continuing with the above equality we get $$ 5x+5y=7x-7y $$
$$ 2x=12y $$ $$ x=6y $$
hence the solutions are $(1,6),(2,12),(3,18)$