Find two positive numbers whose difference is 12 and whose A.M. exceeds the G.M. by 2.
To find the two positive numbers, let's call them 'a' and 'b'.
Given that the difference between the numbers is 12, we can write the equation:
a - b = 12
Next, we are told that the arithmetic mean (A.M.) of the numbers exceeds the geometric mean (G.M.) by 2. The arithmetic mean of two numbers is given by (a + b)/2, and the geometric mean is given by the square root of the product of the two numbers, i.e., √(ab).
So, we can write the equation:
(a + b)/2 - √(ab) = 2
To solve these equations simultaneously, we can use substitution or elimination methods. In this explanation, we'll use the substitution method.
First, let's rewrite the first equation in terms of a:
a = b + 12
Now, substitute this expression for 'a' in the second equation:
((b + 12) + b)/2 - √((b + 12)b) = 2
Simplifying, we have:
(2b + 12)/2 - √(b(b + 12)) = 2
Simplify further:
b + 6 - √(b^2 + 12b) = 2
Rearrange terms:
√(b^2 + 12b) = b + 4
Square both sides of the equation:
b^2 + 12b = (b + 4)^2
Expand the right side:
b^2 + 12b = b^2 + 8b + 16
Now, simplify and rearrange terms:
4b = 16
Divide both sides by 4:
b = 4
Substitute this value of 'b' back into the first equation to find 'a':
a = b + 12
a = 4 + 12
a = 16
Therefore, the two positive numbers are 16 and 4.