The proportional mean of two numbers is 12 more than the smaller of these numbers, and the arithmetic mean

The proportional mean of two numbers is 12 more than the smaller of these numbers, and the arithmetic mean of the same numbers is 24 less than the larger of them. find these numbers.

The proportional mean of two numbers is 12 more than the smaller one, and the arithmetic mean of these numbers is 24 less than the larger number.
Let’s introduce variables.
Let x be a larger number and y a smaller number. Let’s write our equations:
(x * y) ^ (1/2) = y + 12;
(x + y) / 2 + 24 = x;
Multiply the parts of the second equation by 2:
x + y = 48 = 2 * x;
x = y + 48; Substitute x in the first equation:
(y * (y + 48)) ^ (1/2) = y + 12;
(y ^ 2 + 48 * y) ^ (1/2) = y + 12; Squaring both sides:
y ^ 2 + 48 * y = y ^ 2 + 24 * y + 144;
24 * y = 144;
y = 6.
x = 54.
Our numbers are 54 and and 6.