The sum of the cubes of two numbers in the ratio 3:4 is 5824. What is the sum of the two numbers?

Introduction / Context:
This question again involves ratios and cube expressions, but now only two numbers are involved. You are given the ratio of the numbers and the sum of their cubes, and you must find the sum of the numbers themselves. This is typical in algebra and aptitude questions where you use a scaling factor to connect simple ratios to actual numeric values.

Given Data / Assumptions:
- Two numbers are in the ratio 3:4.
- Let the numbers be 3k and 4k, where k is a positive real number.
- The sum of their cubes is 5824.
- Thus, (3k)^3 + (4k)^3 = 5824.
- We must find the sum 3k + 4k.

Concept / Approach:
Write the cubes of the numbers, factor out k^3, and then solve for k^3 by equating to 5824. After finding k, we can compute each number and their sum. This method efficiently transforms an expression involving cubes into a simple equation in a single variable. Once k is known, the sum of the numbers, which is 7k, is easy to calculate.

Step-by-Step Solution:
Step 1: Let the two numbers be 3k and 4k.Step 2: Compute their cubes: (3k)^3 = 27k^3 and (4k)^3 = 64k^3.Step 3: The sum of the cubes is 27k^3 + 64k^3 = 91k^3.Step 4: We are told 91k^3 = 5824.Step 5: Solve for k^3 by dividing both sides by 91: k^3 = 5824 / 91.Step 6: Compute 5824 / 91. Since 91 * 64 = 5824, k^3 = 64.Step 7: The cube root of 64 is 4, because 4^3 = 64. Thus, k = 4.Step 8: The two numbers are 3k = 3 * 4 = 12 and 4k = 4 * 4 = 16.Step 9: The sum of the numbers is 12 + 16 = 28.

Verification / Alternative check:
Check the sum of cubes directly using the numbers 12 and 16. Compute 12^3 = 1728 and 16^3 = 4096. The sum 1728 + 4096 = 5824, which matches the given value. Also, 12 and 16 are in the ratio 3:4, since dividing both by 4 gives 3 and 4. This confirms that our choice of k and the resulting numbers are correct, and their sum of 28 is therefore valid.

Why Other Options Are Wrong:
If the sum were 21, the numbers might be 9 and 12, whose cubes are 729 and 1728, and these do not sum to 5824. A sum of 24 could correspond to 9 and 15, but their cubes 729 and 3375 also do not match 5824. Similarly, sums of 14 or 35 lead to numbers that cannot satisfy the cube sum and ratio conditions simultaneously. Only the sum 28 corresponds to numbers 12 and 16 whose cubes add to 5824 and maintain the 3:4 ratio.

Common Pitfalls:
One mistake is to assume that the sum of cubes gives direct information about the sum of the numbers without considering the ratio. Another is to perform the division 5824 / 91 incorrectly, leading to the wrong value for k^3. Careful calculation of cube values and ratios, and systematic solving for k, ensures the correct answer.

Final Answer:
The sum of the two numbers is 28.