X and Y are two numbers whose A.M. is 41 and G.M. is 9. Which of the following may be a value of X?

Question

X and Y are two numbers whose A.M. is 41 and G.M. is 9.

Which of the following may be a value of X?

1. A. 125
2. B. 81 (Correct answer)
3. C. 49
4. D. 25

Correct Answer

Option B — 81

Detailed Solution & Explanation

The correct answer is 81.

Key Points

• Use the definitions of the two means for the numbers X and Y.
• Arithmetic Mean (A.M.) = (X + Y)/2 = 41 → X + Y = 82.
• Geometric Mean (G.M.) = √(XY) = 9 → XY = 81.
• So X and Y are the roots of the quadratic t² − 82t + 81 = 0, which factorises as (t − 1)(t − 81) = 0, giving t = 1 or t = 81.
• Therefore X can be 81 (and Y = 1).
• for two numbers, AM = (X+Y)/2 and GM = √(XY); they satisfy AM ≥ GM always.
• Given both means, the numbers are recovered as the roots of t² − (sum)t + (product) = 0.

Exam Tip

• here AM = 41 and GM = 9 are far apart, confirming the numbers are very unequal (1 and 81); when AM = GM the two numbers would be identical.