Two numbers x and y are such that their mean proportional is 8 and the third proportional to them is 512. What are the values of x and y?

Introduction / Context:
This aptitude question tests the concept of mean proportional (geometric mean) and third proportional between two numbers. Such questions are common in ratio and proportion chapters in competitive exams. Understanding how to convert verbal statements like “mean proportional is 8” and “third proportional is 512” into algebraic equations is the key. Once the equations are formed, the problem reduces to solving a simple system in x and y.

Given Data / Assumptions:

• x and y are two positive real numbers.
• Their mean proportional (geometric mean) is 8.
• The third proportional to x and y is 512.
• We assume the usual definitions of mean and third proportional from ratio and proportion.

Concept / Approach:
The mean proportional of two numbers x and y is defined as a number m such that x : m = m : y, which gives m^2 = x * y. The third proportional to x and y is a number T such that x : y = y : T, which leads to T = y^2 / x. We will use these definitions to form two equations: one from the mean proportional and one from the third proportional. Solving these equations together gives the required values of x and y.

Step-by-Step Solution:
Mean proportional m = 8 implies m^2 = x * y, so x * y = 8^2 = 64.Third proportional T = 512 means x : y = y : 512, so y^2 / x = 512.From y^2 / x = 512, we get y^2 = 512 * x.We also know x * y = 64, so y = 64 / x.Substitute y = 64 / x into y^2 = 512 * x.(64 / x)^2 = 512 * x.4096 / x^2 = 512 * x.4096 = 512 * x^3.x^3 = 4096 / 512 = 8, so x = 2.Using x * y = 64 gives 2 * y = 64, so y = 32.

Verification / Alternative check:
Check mean proportional: x * y = 2 * 32 = 64, so geometric mean = sqrt(64) = 8, which matches the given condition.Check third proportional: x : y = 2 : 32 = 1 : 16 and y : 512 = 32 : 512 = 1 : 16, so the ratio is preserved. Thus 512 is indeed the third proportional to 2 and 32.

Why Other Options Are Wrong:
Options like “2 and 16”, “4 and 32”, “4 and 16”, or “8 and 8” do not satisfy both conditions simultaneously. For example, if x = 4 and y = 32, the product is 128, whose square root is not 8, so the mean proportional condition fails. Similarly, other pairs do not give y^2 / x equal to 512. Only the pair (2, 32) works for both equations.

Common Pitfalls:
Students sometimes confuse arithmetic mean with geometric mean and incorrectly use (x + y) / 2 = 8 instead of sqrt(x * y) = 8. Another common mistake is misinterpreting the definition of third proportional and writing x^2 / y instead of y^2 / x. Also, some candidates forget to verify both conditions and stop after satisfying only one relation, leading to wrong answers.

Final Answer:
The required numbers are 2 and 32, which satisfy both the mean proportional and the third proportional conditions.