Find the third proportional to the numbers 6 and 12 (that is, 6 : 12 = 12 : x).

Introduction / Context:
This is a direct question on finding the third proportional of two given numbers. It is a basic but important concept in ratio and proportion. Given two numbers a and b, the third proportional c is defined such that a : b = b : c. Here, a = 6 and b = 12, and we must find c that completes the proportion.

Given Data / Assumptions:
• First number a = 6. • Second number b = 12. • Third proportional c = x such that 6 : 12 = 12 : x.

Concept / Approach:
For three numbers a, b and c in continued proportion, we have a : b = b : c. This leads to a / b = b / c. By cross multiplication, we find c = b^2 / a. In this problem, we simply substitute a = 6 and b = 12 into this formula. This approach is straightforward and saves time in exams.

Step-by-Step Solution:
Step 1: Write the proportion: 6 : 12 = 12 : x. Step 2: Convert to a fraction equation: 6 / 12 = 12 / x. Step 3: Simplify the left side: 6 / 12 = 1 / 2. Step 4: So 1 / 2 = 12 / x. Step 5: Cross multiply: x = 12 * 2 = 24.

Verification / Alternative Check:
Check the ratios: 6 : 12 simplifies to 1 : 2. Now compute 12 : 24, which also simplifies to 1 : 2. Since both ratios are equal, the condition 6 : 12 = 12 : 24 is satisfied. This confirms that 24 is the correct third proportional.

Why Other Options Are Wrong:
• 18 and 15 do not satisfy 6 : 12 = 12 : x when substituted for x. • 9 would give 12 : 9 = 4 : 3, which is not the same as 1 : 2.

Common Pitfalls:
Some learners confuse third proportional with mean proportional and use the wrong formula. The mean proportional m between a and b satisfies a : m = m : b, leading to m^2 = a * b, whereas the third proportional c satisfies a : b = b : c, leading to c = b^2 / a. Keeping these definitions distinct avoids confusion.

Final Answer:
The third proportional to 6 and 12 is 24.