Find the two numbers whose AM + GM = 9 and AM: GM = 5/4

Let's denote the two numbers as \(a\) and \(b\). We need to find \(a\) and \(b\) such that the sum of their Arithmetic Mean (AM) and Geometric Mean (GM) equals 9, and the ratio of AM to GM is \( \frac{5}{4} \).

### Step 1: Express AM and GM

The Arithmetic Mean (AM) of \(a\) and \(b\) is:
\[
\text{AM} = \frac{a + b}{2}
\]

The Geometric Mean (GM) of \(a\) and \(b\) is:
\[
\text{GM} = \sqrt{ab}
\]

### Step 2: Set up the equations

According to the problem, we have two conditions:

1. \( \text{AM} + \text{GM} = 9 \)
2. \( \frac{\text{AM}}{\text{GM}} = \frac{5}{4} \)

### Step 3: Substitute AM and GM into the equations

Substitute the expressions for AM and GM into the equations:

1. \( \frac{a + b}{2} + \sqrt{ab} = 9 \)
2. \( \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{5}{4} \)

### Step 4: Simplify the equations

**For the first equation:**

Multiply through by 2:
\[
a + b + 2\sqrt{ab} = 18 \quad \text{(Equation 1)}
\]

**For the second equation:**

Multiply both sides by \(2\sqrt{ab}\):
\[
\frac{a + b}{\sqrt{ab}} = \frac{5}{4} \times 2\sqrt{ab}
\]
\[
\frac{a + b}{\sqrt{ab}} = \frac{5\sqrt{ab}}{2}
\]
Multiply both sides by \(2\sqrt{ab}\) to clear the fraction:
\[
2(a + b) = 5ab \quad \text{(Equation 2)}
\]

### Step 5: Solve the system of equations

We now have two equations:
1. \( a + b + 2\sqrt{ab} = 18 \)
2. \( 2(a + b) = 5ab \)

**Solve Equation 2 for \(a + b\):**
\[
a + b = \frac{5ab}{2}
\]
Substitute this into Equation 1:
\[
\frac{5ab}{2} + 2\sqrt{ab} = 18
\]

Multiply through by 2 to eliminate the fraction:
\[
5ab + 4\sqrt{ab} = 36
\]
Let \(x = \sqrt{ab}\), so \(ab = x^2\). The equation becomes:
\[
5x^2 + 4x - 36 = 0
\]

### Step 6: Solve the quadratic equation

Solve the quadratic equation using the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{4^2 + 4 \times 5 \times 36}}{2 \times 5}
\]
\[
x = \frac{-4 \pm \sqrt{16 + 720}}{10}
\]
\[
x = \frac{-4 \pm \sqrt{736}}{10}
\]
\[
x = \frac{-4 \pm \sqrt{16 \times 46}}{10}
\]
\[
x = \frac{-4 \pm 8.58}{10}
\]

This gives us two possible solutions for \(x\):
\[
x = \frac{-4 + 8.58}{10} \approx 0.458 \quad \text{or} \quad x = \frac{-4 - 8.58}{10} \quad \text{(Not possible as \(x\) must be positive)}
\]

So, \(x = 0.458\), meaning \(ab = x^2 \approx 0.21\).

### Step 7: Solve for \(a\) and \(b\)

Substituting \(x = 0.458\) back into the equation for \(a + b\):
\[
a + b = \frac{5(0.21)}{2} = 0.525
\]

Now, solve the system \(a + b = 0.525\) and \(ab = 0.21\).

This is a quadratic equation:
\[
t^2 - 0.525t + 0.21 = 0
\]

Solve this using the quadratic formula, and you'll find the values of \(a\) and \(b\).

Would you like to see this detailed calculation for the exact values of \(a\) and \(b\)?

Learn how to find two numbers whose Arithmetic Mean (AM) and Geometric Mean (GM) satisfy specific conditions. Detailed step-by-step solution using Arithmetic Mean formula, Geometric Mean formula, and quadratic equation techniques. Suitable for high school students.

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