Finding the Two Numbers from Means

The problem provides the arithmetic mean (AM) and the geometric mean (GM) of two unknown numbers and asks us to find these numbers.

Let the two numbers be \(a\) and \(b\).

Arithmetic Mean Definition

The arithmetic mean of two numbers \(a\) and \(b\) is given by the formula:

\(\text{AM} = \frac{a+b}{2}\)

We are given that the arithmetic mean is 65. So, we can write the first equation:

\(\frac{a+b}{2} = 65\)

Multiplying both sides by 2, we get:

\(a+b = 130 \quad \dots(1)\)

Geometric Mean Definition

The geometric mean of two positive numbers \(a\) and \(b\) is given by the formula:

\(\text{GM} = \sqrt{ab}\)

We are given that the geometric mean is 25. So, we can write the second equation:

\(\sqrt{ab} = 25\)

Squaring both sides of the equation, we get:

\(ab = 25^2\)

\(ab = 625 \quad \dots(2)\)

Solving the System of Equations

We now have a system of two equations with two variables \(a\) and \(b\):

• \(a+b = 130\)
• \(ab = 625\)

We can solve this system. One way is to express one variable in terms of the other from the first equation and substitute it into the second equation.

From equation (1), we can write \(b = 130 - a\).

Substitute this expression for \(b\) into equation (2):

\(a(130 - a) = 625\)

Expand the equation:

\(130a - a^2 = 625\)

Rearrange the terms to form a standard quadratic equation \(ax^2 + bx + c = 0\):

\(a^2 - 130a + 625 = 0\)

Solving the Quadratic Equation

We need to find the values of \(a\) that satisfy this quadratic equation. We can solve this by factoring, completing the square, or using the quadratic formula.

Let's try factoring. We need two numbers that multiply to 625 and add up to -130. Since the product is positive and the sum is negative, both numbers must be negative. However, the original numbers \(a\) and \(b\) must be positive for the geometric mean to be a real number and for the standard AM/GM definition to apply in this context.

Let's think about the roots of the quadratic equation \(x^2 - (a+b)x + ab = 0\). The roots of this equation are precisely the numbers \(a\) and \(b\). In our case, the equation is \(x^2 - 130x + 625 = 0\).

We need two numbers whose sum is 130 and whose product is 625.

Let's list factors of 625:

• 1 and 625 (Sum = 626)
• 5 and 125 (Sum = 130)
• 25 and 25 (Sum = 50)

The pair (5, 125) has a sum of 130 and a product of 625. Therefore, the two numbers are 125 and 5.

Verifying the Numbers

Let's check if these numbers satisfy the given conditions:

Arithmetic Mean: \(\frac{125 + 5}{2} = \frac{130}{2} = 65\). This matches the given AM.

Geometric Mean: \(\sqrt{125 \times 5} = \sqrt{625} = 25\). This matches the given GM.

Thus, the two numbers are 125 and 5.