Question

The sum and difference of the arithmetic mean and the geometric mean of two positive integers are respectively, \(18\) and \(8\). Then the values of the two numbers are

• A. \(12\) and \(24\)
• B. \(2\) and \(24\)
• C. \(6\) and \(20\)
• D. \(8\) and \(18\)
• E. \(1\) and \(25\)

Hint:

For two numbers, once you know the arithmetic mean and geometric mean, first find their sum and product. Then use a quadratic equation to recover the original numbers.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
For two positive numbers, say \(x\) and \(y\), the Arithmetic Mean (AM) is \(\frac{x+y}{2}\) and the Geometric Mean (GM) is \(\sqrt{xy}\). The problem gives us equations based on the sum and difference of these two means.
Step 2: Key Formula or Approach:
Let A be the AM and G be the GM.
We are given:
1) \(A + G = 18\)
2) \(A - G = 8\)
We need to solve this system of linear equations for A and G, and then use the definitions of AM and GM to find the two numbers \(x\) and \(y\).
Step 3: Detailed Explanation:
First, solve for A and G.
Add the two equations:
\[ (A+G) + (A-G) = 18 + 8 \] \[ 2A = 26 \implies A = 13 \] Substitute \(A=13\) into the first equation:
\[ 13 + G = 18 \implies G = 5 \] So, the Arithmetic Mean is 13 and the Geometric Mean is 5.
Now, use the definitions of AM and GM to find the numbers \(x\) and \(y\).
From AM: \(\frac{x+y}{2} = 13 \implies x+y = 26\).
From GM: \(\sqrt{xy} = 5 \implies xy = 5^2 = 25\).
We need to find two numbers whose sum is 26 and whose product is 25. We can form a quadratic equation \(t^2 - (\text{sum of roots})t + (\text{product of roots}) = 0\), where the roots are \(x\) and \(y\).
\[ t^2 - 26t + 25 = 0 \] Factor the quadratic equation:
\[ (t-1)(t-25) = 0 \] The solutions are \(t=1\) and \(t=25\).
So, the two positive integers are 1 and 25.
Step 4: Final Answer:
The values of the two numbers are 1 and 25. Therefore, option (E) is the correct answer.