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:
The arithmetic mean (A.M.) is \(A = \frac{a+b}{2}\) and the geometric mean (G.M.) is \(G = \sqrt{ab}\).
We are given their ratio and need to find the ratio of \((a^2 + b^2)\) to \(ab\).
Step 2: Key Formula or Approach:
Use the given ratio \(\frac{A}{G} = \frac{5}{3}\).
Substitute formulas: \(\frac{a+b}{2\sqrt{ab}} = \frac{5}{3}\).
Square both sides to form algebraic relationships to extract \(a^2 + b^2\).
Step 3: Detailed Explanation:
\[ \frac{a+b}{2\sqrt{ab}} = \frac{5}{3} \implies \frac{a+b}{\sqrt{ab}} = \frac{10}{3} \] Square both sides:
\[ \frac{(a+b)^2}{ab} = \frac{100}{9} \] Expand the numerator:
\[ \frac{a^2 + b^2 + 2ab}{ab} = \frac{100}{9} \] Split the fraction:
\[ \frac{a^2 + b^2}{ab} + 2 = \frac{100}{9} \] Subtract 2 from both sides:
\[ \frac{a^2 + b^2}{ab} = \frac{100}{9} - 2 = \frac{100 - 18}{9} = \frac{82}{9} \] Step 4: Final Answer:
The ratio is 82:9.