The sum of the two numbers is 360, and their ratio is 5: 7, find their numbers.

Let one number be equal to x, the other to y. Then we compose a system of equations:

x + y = 360

x / y = 5/7

Let’s solve the system by the substitution method, expressing one variable through another.
From the second equation it follows that x = 5y / 7. Let’s substitute this expression in the first equation.

y + 5y / 7 = 360

12y / 7 = 360

Find y from the resulting linear equation.

y = 360 * 7/12

y = 210

Substitute the resulting number into the expression x = 5y / 7.

x = 5 * 210/7 = 150.

It follows that the required numbers are 210 and 150.