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 $b$?