Find two numbers such that their mean proportional is 16 and the third proportional to them is 1024. What are the two numbers?

Introduction / Context:
This question is similar to the earlier one involving mean proportional and third proportional, but uses different numerical values. The mean proportional (geometric mean) between two numbers and the third proportional to them are provided. We must compute the original pair of numbers. This tests deeper understanding of classical proportion terminology used in aptitude and algebra questions.

Given Data / Assumptions:

• Let the two numbers be x and y.
• Their mean proportional is 16.
• The third proportional to x and y is 1024.
• We assume x and y are positive real numbers.

Concept / Approach:
The mean proportional m between x and y satisfies x : m = m : y, which implies m^2 = x * y. So x * y = 16^2 = 256. The third proportional T to x and y is defined by x : y = y : T, which implies T = y^2 / x. Given T = 1024, we obtain y^2 / x = 1024. Using x * y = 256 and y^2 / x = 1024, we can solve for x and y systematically.

Step-by-Step Solution:
Let the two numbers be x and y.Mean proportional m = 16 implies x * y = 16^2 = 256.Third proportional T = 1024 implies y^2 / x = 1024.From y^2 / x = 1024, we get y^2 = 1024 * x.From x * y = 256, we can write y = 256 / x.Substitute y = 256 / x into y^2 = 1024 * x.(256 / x)^2 = 1024 * x.65536 / x^2 = 1024 * x.Multiply both sides by x^2: 65536 = 1024 * x^3.x^3 = 65536 / 1024 = 64.So x = cube root of 64 = 4.Now use x * y = 256: 4 * y = 256, so y = 64.

Verification / Alternative check:
Check mean proportional: x * y = 4 * 64 = 256, geometric mean = sqrt(256) = 16, which matches the given mean proportional.Check third proportional: x : y = 4 : 64 = 1 : 16 and y : T must also be 1 : 16.With T = 1024, y : T = 64 : 1024 = 1 : 16, so 1024 is indeed the third proportional to 4 and 64.

Why Other Options Are Wrong:
Pairs like (4, 32), (8, 64) or (8, 32) do not satisfy both conditions simultaneously. For instance, (4, 32) gives product 128 and geometric mean sqrt(128), which is not 16. Similarly, other options fail either the mean proportional condition or the third proportional condition. Only 4 and 64 satisfy both.

Common Pitfalls:
Students sometimes confuse arithmetic mean with geometric mean and use (x + y) / 2 = 16 instead of x * y = 256. Another mistake is to misinterpret the definition of third proportional and write x^2 / y instead of y^2 / x. Consistently using the correct definitions and double checking each condition after solving helps to avoid these errors.

Final Answer:
The two numbers are 4 and 64.