Sum of Numbers Setup

Let the two positive numbers be represented by \(x\) and \(y\). According to the problem statement:

• The sum of the two numbers is 40: \(x + y = 40\).
• The Geometric Mean (GM) is 20% lower than the Arithmetic Mean (AM).

We need to find the difference between the two numbers, which is \(|x - y|\).

Arithmetic Mean (AM) Calculation

The Arithmetic Mean (AM) of two numbers \(x\) and \(y\) is calculated as:

AM = \(\frac{x+y}{2}\)

Substituting the given sum (\(x + y = 40\)):

AM = \(\frac{40}{2} = 20\)

Geometric Mean (GM) Calculation

The problem states that the GM is 20% lower than the AM.

This can be written as:

GM = AM - (20\% \text{ of AM})

GM = AM \times (1 - 0.20)

GM = AM \times 0.80

Substituting the calculated AM value:

GM = \(20 \times 0.80 = 16\)

The Geometric Mean (GM) of two numbers \(x\) and \(y\) is calculated as:

GM = \(\sqrt{xy}\)

Now, we can find the product \(xy\):

\(\sqrt{xy} = 16\)

Squaring both sides:

\(xy = 16^2 = 256\)

Difference Between the Two Numbers

We now have two key pieces of information:

1. Sum: \(x + y = 40\)
2. Product: \(xy = 256\)

To find the difference \(|x - y|\), we can use the algebraic identity:

\((x - y)^2 = (x + y)^2 - 4xy\)

Substitute the values of the sum and product:

\((x - y)^2 = (40)^2 - 4(256)\)

\((x - y)^2 = 1600 - 1024\)

\((x - y)^2 = 576\)

To find the difference, take the square root of both sides:

\(|x - y| = \sqrt{576}\)

Calculating the square root:

\(|x - y| = 24\)

Conclusion

The difference between the two positive numbers is 24.