Question

The A.M. and G.M. of two positive real numbers $a$ and $b$ ($a > b$) are $A$ and $G$ respectively. If $A:G = 5:3$, then $(a^2 + b^2) : ab = $

• A. 83:9
• B. 82:9
• C. 83:6
• D. 82:7
• E. 9:1

Hint:

When asked for ratios of squares and products like $(a^2+b^2)/ab$, always start by squaring the expression for $(a+b)/\sqrt{ab}$ or $(a-b)/\sqrt{ab}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Arithmetic Mean (A.M.) is $\frac{a+b}{2}$ and Geometric Mean (G.M.) is $\sqrt{ab}$. The ratio of these means can be related to the algebraic expression requested.
Key Formula or Approach:
Given $\frac{A}{G} = \frac{5}{3} \implies \frac{a+b}{2\sqrt{ab}} = \frac{5}{3}$.

Step 2: Detailed Explanation:
1. Simplify the given ratio:
\[ \frac{a+b}{\sqrt{ab}} = \frac{10}{3} \]
2. Square both sides to eliminate the square root:
\[ \frac{(a+b)^2}{ab} = \left(\frac{10}{3}\right)^2 = \frac{100}{9} \]
3. Expand the numerator:
\[ \frac{a^2 + b^2 + 2ab}{ab} = \frac{100}{9} \]
4. Separate the terms:
\[ \frac{a^2 + b^2}{ab} + \frac{2ab}{ab} = \frac{100}{9} \]
\[ \frac{a^2 + b^2}{ab} + 2 = \frac{100}{9} \]
5. Solve for the desired ratio:
\[ \frac{a^2 + b^2}{ab} = \frac{100}{9} - 2 = \frac{100 - 18}{9} = \frac{82}{9} \]

Step 3: Final Answer:
The ratio $(a^2 + b^2) : ab$ is 82:9.