Two positive numbers x and y are such that their mean proportional (geometric mean) is 16 and the third proportional to x and y is 128. What are the values of x and y?

Introduction / Context:
This problem involves the classical concepts of mean proportional (geometric mean) and third proportional between two numbers. These terms often appear in ratio and proportion topics in aptitude and school mathematics. The question gives the value of the geometric mean and the third proportional, and asks for the original pair of numbers x and y that satisfy both conditions.

Given Data / Assumptions:

• Mean proportional (geometric mean) of x and y is 16.
• Third proportional to x and y is 128.
• x and y are positive real numbers.
• By definition, if a : b = b : c, then b is the mean proportional and c is the third proportional.

Concept / Approach:
For two positive numbers x and y, their geometric mean is sqrt(x * y). So the first condition gives x * y = 16^2 = 256. The third proportional condition means x : y = y : 128. This gives a second relationship between x and y. Using these two equations together, we can solve for x and y. Knowledge of these special proportional definitions is the key to solving the question quickly.

Step-by-Step Solution:
From the geometric mean condition: sqrt(x * y) = 16. Squaring both sides gives x * y = 256. (Equation 1) From the third proportional condition: x : y = y : 128. This means x / y = y / 128. Cross multiply to get 128 * x = y^2. (Equation 2) From Equation 1, x = 256 / y. Substitute into Equation 2: 128 * (256 / y) = y^2. This simplifies to (128 * 256) / y = y^2. So y^3 = 128 * 256. Compute 128 * 256 = 2^7 * 2^8 = 2^15 = 32768. Hence y^3 = 32768, so y = cube root of 32768 = 32. From x * y = 256, x = 256 / 32 = 8. Therefore, x = 8 and y = 32.

Verification / Alternative check:
Check the geometric mean: sqrt(8 * 32) = sqrt(256) = 16, which matches the given mean proportional. Check the third proportional condition: x : y = 8 : 32 simplifies to 1 : 4. Also, y : 128 = 32 : 128, which simplifies to 1 : 4. Therefore 8 : 32 = 32 : 128, confirming that 128 is indeed the third proportional to 8 and 32. Both conditions are satisfied.

Why Other Options Are Wrong:
Pairs such as 8 and 16 or 16 and 32 do not produce both a geometric mean of 16 and a third proportional equal to 128 when tested. For example, 8 and 16 give geometric mean sqrt(128), not 16; 16 and 16 give third proportional 16, not 128.

Common Pitfalls:
Learners often confuse arithmetic mean with geometric mean and incorrectly use (x + y) / 2 instead of sqrt(x * y). Some also misremember the definition of third proportional, mistakenly using x : y = 128 : y rather than x : y = y : 128. Careful recall of definitions and systematic use of algebraic equations ensures correct results.

Final Answer:
The two numbers are x = 8 and y = 32.