Question

If the mean of the observations 6, 16, \( (K + 26) \), \( (2K - 3) \), and 10 is 14, then the value of \( \frac{K^2 + 1}{2} \) is:

• A. 25
• B. 18.5
• C. 13
• D. 8.5

Hint:

For average problems, always multiply both sides of the equation by the number of terms to simplify the calculation. Then solve for the unknown.

Answer 1

The Correct Option is C

We are given that the mean of the observations is 14. The sum of the observations is: \[ \frac{6 + 16 + (K + 26) + (2K - 3) + 10}{5} = 14 \] Simplifying the numerator: \[ 6 + 16 + K + 26 + 2K - 3 + 10 = 55 + 3K \] Thus, the equation becomes: \[ \frac{55 + 3K}{5} = 14 \] Multiplying both sides by 5: \[ 55 + 3K = 70 \] Solving for \( K \): \[ 3K = 15 \quad \Rightarrow \quad K = 5 \] Now, substitute \( K = 5 \) into the expression \( \frac{K^2 + 1}{2} \): \[ \frac{K^2 + 1}{2} = \frac{5^2 + 1}{2} = \frac{25 + 1}{2} = \frac{26}{2} = 13 \] Thus, the correct answer is \( \boxed{13} \).