Ratio between two numbers is 6:7 and the difference between them is 10. What are the two numbers?

Question

I know the numbers are $60$ and $70$ but I got that by trial and error. Is there some other more logical way to do this problem?

Answer 1

Call your two numbers $x$ and $y$, with $x$ being the bigger.

Then what you know can be expressed as: $$ \frac{y}{x} = \frac{6}{7} $$ and $$ x-y=10 $$ That is a fairly easy system of equations to solve.

Answer 2

Call your two numbers $x$ and $y$. We have

$$x:y = 6:7 \implies x = \frac67y\\ \begin{align*} \left|x-y\right| &= 10\\ \left|\frac67y-y\right| &= 10\\ \left|\frac67-1\right|\left|y\right| &= 10\\ \left|y\right| &= 70\\ y &= 70 \text{ or } -70\\ x &= 60 \text{ or } -60 \text{ respectively} \end{align*}$$

Answer 3

If $~\dfrac xy=\dfrac67~$ then $x=6k$ and $y=7k.~$ Indeed, $~\dfrac{6k}{7k}=\dfrac67.~$ Now, $y-x=7k-6k=k=10$.