Third proportional to the numbers 10 and 20 is defined as the number x such that 10 : 20 = 20 : x. What is the value of x?

Introduction:
This question tests your understanding of the concept of third proportional in ratio and proportion. Finding the third proportional is a very common topic in quantitative aptitude.

Given Data / Assumptions:
First number = 10. Second number = 20. Third proportional = x, such that 10 : 20 = 20 : x.

Concept / Approach:
If a, b and c are such that a : b = b : c, then c is called the third proportional to a and b. This ratio equality can be written as: a / b = b / c. From this we get: c = (b * b) / a. We apply this formula directly with a = 10 and b = 20.

Step-by-Step Solution:
Step 1: Write the proportion condition. 10 : 20 = 20 : x. Step 2: Convert to fraction form. 10 / 20 = 20 / x. Step 3: Solve for x by cross multiplication. 10 * x = 20 * 20. 10x = 400. x = 400 / 10 = 40.

Verification / Alternative Check:
Check the ratio: 10 : 20 = 1 : 2. For 20 : 40, we also get 1 : 2. The ratios match, confirming that 40 is indeed the correct third proportional to 10 and 20.

Why Other Options Are Wrong:
30, 25, 20, 50: Substituting any of these values for x makes 20 : x different from 10 : 20. The ratios no longer match, so they cannot be third proportionals.

Common Pitfalls:
Some learners mistakenly take the arithmetic mean or the geometric mean instead of using the proportion definition. Others may incorrectly write a * c = b instead of a : b = b : c. Always recall that third proportional c satisfies c = b² / a.

Final Answer:
The third proportional to 10 and 20 is 40.