Find the third proportional to the two numbers 9 and 24, that is, the number which forms a proportion 9 : 24 = 24 : x.

Introduction / Context:
This question tests the concept of third proportional in ratio and proportion. The idea of third proportional is fundamental in many algebra and aptitude problems, where a proportion of the form a : b = b : c is used. Here, the numbers 9 and 24 are given and we must find the third proportional that completes the proportion with 24 as the middle term.

Given Data / Assumptions:

• The first number is 9.• The second number is 24.• We need to find the third proportional x such that 9 : 24 = 24 : x.• All numbers are positive real numbers.

Concept / Approach:
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. In such a proportion, we have a / b = b / c. Cross multiplication then gives a * c = b * b. From this relation, c can be found as c = (b^2) / a. Applying this directly to 9 and 24 allows us to compute the third proportional quickly and accurately.

Step-by-Step Solution:
Step 1: Let the third proportional be x so that 9 : 24 = 24 : x.Step 2: Write this proportion in fractional form: 9 / 24 = 24 / x.Step 3: Cross multiply to eliminate the fractions: 9 * x = 24 * 24.Step 4: Compute 24 * 24 = 576.Step 5: So 9 * x = 576.Step 6: Solve for x by dividing both sides by 9: x = 576 / 9.Step 7: Perform the division: 576 / 9 = 64.Step 8: Therefore, the third proportional to 9 and 24 is 64.

Verification / Alternative check:
To verify, check that 9 : 24 = 24 : 64. First, simplify 9 : 24 by dividing both numbers by 3, giving 3 : 8. Next, simplify 24 : 64 by dividing both numbers by 8, giving 3 : 8. Since both simplified ratios are equal, 9 : 24 = 24 : 64 holds true. This confirms that 64 is indeed the correct third proportional.

Why Other Options Are Wrong:
• 39: Plugging 39 in place of x would give 9 / 24 ≠ 24 / 39, so the proportion is not satisfied.• 48: If x were 48, then 24 : 48 simplifies to 1 : 2, whereas 9 : 24 simplifies to 3 : 8, which are not equal ratios.• 72: If x were 72, the ratio 24 : 72 simplifies to 1 : 3, again not matching 9 : 24.

Common Pitfalls:
Students sometimes confuse the concept of mean proportional (or geometric mean) with third proportional or mistakenly use the formula for fourth proportional. Another frequent issue is to reverse the position of numbers in the proportion and write 24 : 9 = 24 : x, which changes the meaning completely. Miscalculating 24 squared or dividing 576 by 9 incorrectly can also lead to wrong results. Being clear about the definition of third proportional and using the formula c = b^2 / a helps avoid such errors.

Final Answer:
The third proportional to 9 and 24 is 64.