The difference between two numbers refers to their product as 1:12, the sum of the numbers is 3 times

The difference between two numbers refers to their product as 1:12, the sum of the numbers is 3 times their difference. find these numbers.

Let us denote the required numbers by x and y.
According to the condition of the problem, the difference between these two numbers refers to their product as 1:12, therefore, we can write the following ratio:
(x – y) / (x * y) = 1/12.
It is also known that the sum of these two numbers is 3 times their difference, therefore, we can write the following ratio:
x + y = 3 * (x – y).
We solve the resulting system of equations.
Simplifying the second ratio, we get:
x + y = 3x – 3y;
3x – x = 3y + y;
2x = 4y;
x = 4y / 2;
x = 2y.
Substituting the resulting value of x and the ratio (x – y) / (x * y) = 1/12, we get:
(2y – y) / (2y * y) = 1/12;
y / (2y * y) = 1/12;
1 / 2y = 1/12;
2y = 12;
y = 12/2;
y = 6.
Knowing y, we find x:
x = 2y = 2 * 6 = 12.
Answer: the required numbers are 12 and 6.