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:
- Sum: \(x + y = 40\)
- 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.
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