Two numbers are in the ratio of $7:6$. If the difference between their squares is $637$, then the numbers are:

Let the two numbers be $7x$ and $6x$.According to the question, the difference between their squares is $637$:
$$(7x{)}^2-(6x{)}^2=637$$
$$49x^2-36x^2=637$$
$$13x^2=637$$
$$x^2=\frac{637}{13}$$
$$x^2=49$$
$$x=7$$
Now, substitute $x=7$ back into our expressions for the two numbers:First number = $7x=7\times 7=49$ Second number = $6x=6\times 7=42$ Thus, the numbers are $49$ and $42$.