What is the third proportional to the numbers 9 and 45, that is, the value x such that 9 : 45 = 45 : x?

Introduction / Context:
This question asks for the third proportional to two given numbers. 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. This is a standard concept in ratio and proportion that often appears in aptitude questions. Here we are given a = 9 and b = 45 and asked to find c.

Given Data / Assumptions:
- First number a = 9.
- Second number b = 45.
- We want c such that 9 : 45 = 45 : c.
- All numbers are positive real numbers.

Concept / Approach:
From the definition, for a, b, c to be in continued proportion we must have a/b = b/c. This can be rearranged to c = b^2 / a. Therefore, once a and b are known, we compute c by squaring the second number and dividing by the first. This method is a quick way to obtain the third proportional without explicitly dealing with the full ratio fraction.

Step-by-Step Solution:
Step 1: Use the proportion 9 : 45 = 45 : x where x is the third proportional. Step 2: Write it as a fraction: 9 / 45 = 45 / x. Step 3: Cross multiply to get 9x = 45 * 45. Step 4: Compute 45 * 45 = 2,025. Step 5: Solve 9x = 2,025 to get x = 2,025 / 9. Step 6: Compute 2,025 / 9 = 225. Step 7: Therefore, the third proportional to 9 and 45 is 225.

Verification / Alternative check:
As a check, verify that the ratio 9 : 45 equals the ratio 45 : 225. Simplify 9 : 45 to 1 : 5 and 45 : 225 also simplifies to 1 : 5 since both numerator and denominator are divided by 45. Because the ratios match, we confirm that 225 maintains the correct proportional relationship and is indeed the third proportional.

Why Other Options Are Wrong:
405 and 81 do not satisfy the condition 9 : 45 = 45 : x when substituted for x. For example, 45 : 405 simplifies to 1 : 9, not 1 : 5. The value 5 is far too small and completely breaks the proportional sequence. Only 225 makes both ratios equal, so it is the unique correct answer by the definition of third proportional.

Common Pitfalls:
A common error is to confuse third proportional with mean proportional, which is related to geometric mean, or to improperly square the wrong number. Some learners also forget to simplify the final ratio when checking their result, leading them to think a wrong number might also work. Remembering the formula c = b^2 / a is a fast and reliable way to handle such questions.

Final Answer:
The third proportional to 9 and 45 is 225.