Question

Arithmetic mean & Geometric mean of 2 numbers \( a \) & \( b \) in the ratio 5:3. Find \( \frac{a^2 + b^2}{ab} \).

Hint:

To solve such problems, use the relationship between the arithmetic and geometric means to create a system of equations.

Answer 1

Step 1: Use the formula for arithmetic mean and geometric mean.
The arithmetic mean (AM) and geometric mean (GM) of two numbers \( a \) and \( b \) are given by: \[ \text{AM} = \frac{a + b}{2}, \quad \text{GM} = \sqrt{ab} \] We are given that the ratio of AM to GM is \( \frac{5}{3} \), i.e.: \[ \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{5}{3} \]
Step 2: Simplify the equation.
Simplifying the equation: \[ \frac{a + b}{2\sqrt{ab}} = \frac{5}{3} \] Multiplying both sides by \( 2\sqrt{ab} \): \[ a + b = \frac{10}{3} \sqrt{ab} \]
Step 3: Square both sides to eliminate the square root.
Squaring both sides of the equation: \[ (a + b)^2 = \left( \frac{10}{3} \right)^2 ab \] This simplifies to: \[ a^2 + 2ab + b^2 = \frac{100}{9} ab \]
Step 4: Express \( \frac{a^2 + b^2}{ab} \).
Now, we need to find \( \frac{a^2 + b^2}{ab} \). From the above equation: \[ a^2 + b^2 = \frac{100}{9} ab - 2ab = \left( \frac{100}{9} - 2 \right) ab = \frac{100}{9} - \frac{18}{9} ab = \frac{82}{9} ab \] Thus, \[ \frac{a^2 + b^2}{ab} = \frac{82}{9} \]