What is the third proportional to the pair of numbers 9 and 15?

Introduction / Context:
This question tests the concept of third proportional between two numbers, which appears frequently in ratio and proportion chapters. Understanding proportional relationships is important in many aptitude topics, including similarity in geometry, financial comparisons, and mixture problems. The problem itself is short, but it checks whether the student remembers the specific definition of the third proportional and can apply the formula correctly.

Given Data / Assumptions:

First number a = 9.
Second number b = 15.
We are asked to find the third proportional c such that a : b = b : c.
All numbers are positive real numbers and normal arithmetic rules apply.

Concept / Approach:
By definition, if a, b, c are in continued proportion, then a : b = b : c. This gives a proportion equation a / b = b / c. Cross multiplication yields a * c = b * b, so c = b^2 / a. Thus to find the third proportional, we simply square the second number and divide by the first number. This is a direct and very fast method that avoids unnecessary steps.

Step-by-Step Solution:
Given a = 9 and b = 15. For third proportional c, we require 9 : 15 = 15 : c. Write this as 9 / 15 = 15 / c. Cross multiply to get 9 * c = 15 * 15. So 9 * c = 225. Therefore c = 225 / 9 = 25.

Verification / Alternative check:
Check the ratio condition directly. With c = 25, first ratio 9:15 simplifies to 3:5 when both terms are divided by 3. Second ratio 15:25 simplifies to 3:5 when both terms are divided by 5. Since both ratios reduce to 3:5, the condition 9:15 = 15:25 is satisfied, confirming that 25 is indeed the third proportional.

Why Other Options Are Wrong:
Option 30 would give a second ratio 15:30, which simplifies to 1:2, not equal to 9:15 which simplifies to 3:5.
Option 27 would yield a ratio 15:27 equal to 5:9, again not equal to 3:5, so it does not satisfy the proportionality condition.
Option 36 leads to 15:36 which simplifies to 5:12, again different from the required ratio 3:5, so this choice is incorrect.

Common Pitfalls:
Some learners mistakenly compute c as a * b instead of b^2 / a, confusing the definition of continued proportion.
Others mix up the idea of mean proportional (which involves square roots) with third proportional, which uses the square of the middle term.
Another frequent error is to perform addition or subtraction on the numbers instead of maintaining the ratio equality through cross multiplication.

Final Answer:
The third proportional to 9 and 15 is 25.