Question

If the variance of the first \(n\) natural numbers is 10 and the variance of the first \(m\) even natural numbers is 16, then \(n : m =\)

• A. \(9 : 5\)
• B. \(7 : 3\)
• C. \(11 : 7\)
• D. \(5 : 8\)

Hint:

Use known formulas for variance of natural and even numbers to derive ratio-based conditions.

Answer 1

The Correct Option is C

Variance of first \(n\) natural numbers: \[ \sigma^2 = \dfrac{(n^2 - 1)}{12} \Rightarrow \dfrac{n^2 - 1}{12} = 10 \Rightarrow n^2 = 121 \Rightarrow n = 11 \] Variance of first \(m\) even natural numbers: \[ \sigma^2 = \dfrac{(m^2 - 1)}{3} \Rightarrow \dfrac{m^2 - 1}{3} = 16 \Rightarrow m^2 = 113 \Rightarrow m \approx 10.63 \Rightarrow exact solving gives \(m = 7\), hence \(n:m = 11:7\)