Question

If two positive numbers are in the ratio $ 3 + 2\sqrt{2} : 3 - 2\sqrt{2} $, then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is:

• A. \( 3 : 4 \)
• B. \( 6 : 1 \)
• C. \( 3 : 2 \)
• D. \( 3 : 1 \)

Hint:

When dealing with ratios of numbers, simplify the ratio first before proceeding with calculating the means. This can make the problem much easier to solve.

Answer 1

The Correct Option is D

Let the two numbers be \( a \) and \( b \). We are given their ratio: \[ \frac{a}{b} = \frac{3 + 2\sqrt{2}}{3 - 2\sqrt{2}}. \] To simplify this ratio, multiply both the numerator and the denominator by the conjugate of the denominator: \[ \frac{a}{b} = \frac{(3 + 2\sqrt{2})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})}. \] Simplifying the denominator: \[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1. \] Now, simplifying the numerator: \[ (3 + 2\sqrt{2})^2 = 3^2 + 2 \cdot 3 \cdot 2\sqrt{2} + (2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2}. \] Thus, we have: \[ \frac{a}{b} = 17 + 12\sqrt{2}. \] Now, let's find the arithmetic mean (A.M) and geometric mean (G.M) of \( a \) and \( b \). 
1. The A.M of \( a \) and \( b \) is: \[ A.M = \frac{a + b}{2}. \] 
2. The G.M of \( a \) and \( b \) is: \[ G.M = \sqrt{ab}. \] The ratio of A.M to G.M is: \[ \frac{A.M}{G.M} = \frac{\frac{a + b}{2}}{\sqrt{ab}} = 3 : 1. \] Thus, the correct answer is \( 3 : 1 \).