Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. What is the value of the smallest number?

Introduction / Context:
This problem combines the concept of ratios with cubic expressions. The three numbers are in a simple ratio, but instead of their sum, you are given the sum of their cubes. From that information, you must deduce the actual values of the numbers and then identify the smallest one. This is a good example of how ratio problems can lead to polynomial equations in the scaling factor.

Given Data / Assumptions:
- Three numbers are in the ratio 1:2:3.
- Let the common multiplying factor be k, so the numbers are k, 2k, and 3k.
- The sum of the cubes of these three numbers is 4500.
- Mathematically, k^3 + (2k)^3 + (3k)^3 = 4500.
- We need to find the smallest of the three numbers.

Concept / Approach:
Since the numbers are scaled versions of 1, 2, and 3, their cubes will be scaled versions of 1^3, 2^3, and 3^3. By factoring out k^3 from the sum of cubes, the equation simplifies significantly and we can solve for k^3. Once k is determined, the smallest number is simply k. This approach avoids guessing and makes full use of the ratio information.

Step-by-Step Solution:
Step 1: Let the three numbers be k, 2k, and 3k.Step 2: Compute their cubes: k^3, (2k)^3 = 8k^3, and (3k)^3 = 27k^3.Step 3: The sum of their cubes is k^3 + 8k^3 + 27k^3 = 36k^3.Step 4: We are told that this sum equals 4500, so 36k^3 = 4500.Step 5: Solve for k^3 by dividing both sides by 36: k^3 = 4500 / 36.Step 6: Simplify 4500 / 36. Divide numerator and denominator by 4 to get 1125 / 9.Step 7: Compute 1125 / 9 = 125.Step 8: Therefore, k^3 = 125, which implies k = 5 because 5^3 = 125.Step 9: The three numbers are k = 5, 2k = 10, and 3k = 15.Step 10: Among these, the smallest number is 5.

Verification / Alternative check:
Check the cubes of 5, 10, and 15. We have 5^3 = 125, 10^3 = 1000, and 15^3 = 3375. Their sum is 125 + 1000 + 3375 = 4500, which matches the given total. This confirms that k = 5 is correct and that the numbers 5, 10, and 15 are indeed the original numbers in the ratio 1:2:3 with the correct cube sum.

Why Other Options Are Wrong:
If the smallest number were 4, the three numbers would be 4, 8, and 12, and their cubes would sum to 64 + 512 + 1728 = 2304, not 4500. If the smallest number were 6, the numbers 6, 12, and 18 would have cubes 216, 1728, and 5832, whose sum is far larger than 4500. Similarly, smallest numbers 8 or 10 lead to cube sums that do not match 4500. Only 5 produces the correct total cube sum.

Common Pitfalls:
A typical error is to assume that the sum of cubes being 4500 implies the sum of the numbers is some simple value, and then misapply a ratio directly to that sum. Others may forget to factor out k^3 and instead attempt to manipulate the equation in a less organised way. Always express the numbers in terms of the common factor and carefully compute the cubes and their sum.

Final Answer:
The smallest of the three numbers is 5.