What is the third proportional to the numbers 10 and 20?

Introduction / Context:
This question is about the concept of third proportional between two numbers, which is a common topic in ratio and proportion. Understanding proportionality is important for solving many geometry, algebra and word problems. The question asks for a single number that completes a specific proportional relationship with 10 and 20.

Given Data / Assumptions:

• First number = 10.
• Second number = 20.
• We need to find the third proportional to 10 and 20.

Concept / Approach:
If a, b and c are in proportion such that a : b = b : c, then c is called the third proportional to a and b. From a : b = b : c, we can write a / b = b / c and then cross-multiply to get a * c = b^2. Therefore, c = b^2 / a. This formula can be applied directly to find the third proportional when the first two numbers are known.

Step-by-Step Solution:
1) Let a = 10 and b = 20. 2) We want c such that 10 : 20 = 20 : c. 3) Write the proportion in fraction form: 10 / 20 = 20 / c. 4) Simplify 10 / 20 to 1 / 2. 5) So 1 / 2 = 20 / c. 6) Cross-multiply to get c = 20 * 2 = 40. 7) Therefore, the third proportional to 10 and 20 is 40.
Verification / Alternative check:
We can verify by checking the original proportional relation. If a = 10, b = 20 and c = 40, then 10 : 20 = 1 : 2 and 20 : 40 = 1 : 2 as well. Since both ratios are equal, 10 : 20 = 20 : 40, confirming that 40 is indeed the correct third proportional. Another way is to use the formula c = b^2 / a = 20^2 / 10 = 400 / 10 = 40, which gives the same result.

Why Other Options Are Wrong:
Option A (30), option B (25), option C (50) and option E (20) do not satisfy the condition 10 : 20 = 20 : c. For example, with c = 50, 20 : 50 simplifies to 2 : 5, which is not the same as 10 : 20 = 1 : 2. Similarly, none of the other proposed values make the two ratios identical. Only c = 40 ensures equality of the two ratios, as required for a third proportional.

Common Pitfalls:
Learners often confuse the definition of third proportional with mean proportional (geometric mean) or with the idea of a simple arithmetic progression. Some also mistakenly use c = a * b instead of c = b^2 / a. Carefully remembering that in a : b = b : c, the middle term b appears in both ratios is crucial to setting up and solving the correct equation for the third proportional.

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