Understanding the Averages
The arithmetic mean (AM) and geometric mean (GM) of two items can help us find the items and their difference. Let's denote the two items as \(x\) and \(y\).
Given Data
- Arithmetic Mean:
\( AM = \frac{x + y}{2} = 12.5 \)
- Geometric Mean:
\( GM = \sqrt{xy} = 10 \)
Formulating Equations
From the arithmetic mean, we can derive:
- \( x + y = 2 \times 12.5 = 25 \)
From the geometric mean, we can derive:
- \( xy = 10^2 = 100 \)
Now we have two equations:
1. \( x + y = 25 \)
2. \( xy = 100 \)
Finding the Items
We can solve these equations simultaneously. We can express \(y\) in terms of \(x\):
- \( y = 25 - x \)
Substituting this into the second equation:
- \( x(25 - x) = 100 \)
- \( 25x - x^2 = 100 \)
- Rearranging gives:
\( x^2 - 25x + 100 = 0 \)
Using the Quadratic Formula
To find \(x\), we apply the quadratic formula:
- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Here, \( a = 1, b = -25, c = 100 \)
Calculating the discriminant:
- \( b^2 - 4ac = 625 - 400 = 225 \)
- So, \( x = \frac{25 \pm 15}{2} \)
This gives us two solutions for \(x\):
1. \( x = 20 \)
2. \( x = 5 \)
Calculating the Difference
Using \( x = 20 \) and \( y = 5 \):
- Difference:
\( |x - y| = |20 - 5| = 15 \)
Thus, the difference between the two items is 15.