What is the third proportional to the numbers 9 and 12?

Introduction / Context:
The concept of third proportional is a classic topic in ratio and proportion. It helps in understanding proportional relationships where three numbers satisfy a specific ratio condition and is often tested in quantitative aptitude sections of competitive exams.

Given Data / Assumptions:

• We are given two numbers: 9 and 12.
• We must find the third proportional to 9 and 12.

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. In equation form, a / b = b / c, which implies a * c = b * b. So c = b^2 / a. We will substitute a = 9 and b = 12 into this formula to find c.

Step-by-Step Solution:
Step 1: Let the required third proportional be x. Step 2: By definition, 9 : 12 = 12 : x. Step 3: Convert this ratio statement into a fraction equation: 9 / 12 = 12 / x. Step 4: Cross multiply to get 9x = 12 * 12. Step 5: Compute 12 * 12 = 144. Step 6: So 9x = 144. Therefore x = 144 / 9. Step 7: Simplify 144 / 9 = 16. Step 8: Hence the third proportional is 16.

Verification / Alternative check:
Check whether the ratio 9 : 12 equals 12 : 16. Compute 9 / 12 = 0.75 and 12 / 16 = 0.75. Since they are equal, the condition a : b = b : c is satisfied. This confirms that 16 is indeed the correct third proportional to 9 and 12.

Why Other Options Are Wrong:
Values like 14, 18, 20 or 24 do not satisfy the proportional relationship 9 : 12 = 12 : x. For each of these, 12 / x gives a fraction that is not equal to 9 / 12, so they cannot be the third proportional. For example, 12 / 18 = 2 / 3 which is different from 9 / 12 = 3 / 4.

Common Pitfalls:
Students sometimes confuse third proportional with mean proportional (geometric mean). For a third proportional, the equation is 9 : 12 = 12 : x, not 9 : x = x : 12. Another error is to attempt to subtract numbers or take an average instead of using the correct proportion formula. Always remember that third proportional uses the pattern a : b = b : c which translates into b^2 = a * c.

Final Answer:
The third proportional to 9 and 12 is 16.