The third proportional to 10 and 20 is the number x such that 10 : 20 = 20 : x. What is the value of this third proportional?

Introduction / Context:
This question is about proportion and specifically the concept of a third proportional. It tests your understanding of ratios and how to convert a statement such as “a is to b as b is to c” into an algebraic equation that you can solve for the unknown value c.

Given Data / Assumptions:

• We are given two numbers: 10 and 20.

• A third proportional x is defined so that 10 : 20 = 20 : x.

• All numbers are positive real numbers.

• We need to find the value of x that satisfies this proportion.

Concept / Approach:
If a : b = b : c, then b is called the mean proportional and c is called the third proportional to a and b. The ratio a : b = b : c can be rewritten as a / b = b / c. From this, we can cross multiply to form an equation a * c = b * b. Solving this equation for c gives c = b^2 / a. Here a = 10, b = 20 and c = x, so we apply this formula directly.

Step-by-Step Solution:
Given 10 : 20 = 20 : x. Write it as a fraction: 10 / 20 = 20 / x. Simplify 10 / 20 to 1 / 2. So, 1 / 2 = 20 / x. Cross multiply: 1 * x = 2 * 20. x = 40. Alternatively, use the formula c = b^2 / a = 20^2 / 10 = 400 / 10 = 40.

Verification / Alternative check:
Check the proportion: 10 : 20 equals 1 : 2. Now check 20 : 40, which is also 1 : 2. Since both ratios are equal, the proportion 10 : 20 = 20 : 40 holds true. Therefore, 40 correctly acts as the third proportional to 10 and 20 according to the definition.

Why Other Options Are Wrong:
If x were 30, then 20 : 30 simplifies to 2 : 3, not 1 : 2, so the ratios would not match. For x = 25, 20 : 25 = 4 : 5, and for x = 20, 20 : 20 = 1 : 1. None of these equal 10 : 20, which is 1 : 2. Hence those values do not satisfy the proportional relationship required for a third proportional.

Common Pitfalls:
A common mistake is to confuse third proportional with mean proportional (geometric mean), where the relationship is different. Another error is to forget to cross multiply and simplify ratios before solving. To avoid confusion, always start by writing the ratio as a fraction and then apply cross multiplication carefully.

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