Difficulty: Medium
Correct Answer: 5
Explanation:
Introduction:
This question tests your understanding of ratios combined with powers of numbers, specifically cubes. It is a good example of how ratio-based expressions can be translated into algebraic equations and then solved using basic arithmetic and cube properties.
Given Data / Assumptions:
Concept / Approach:
If three numbers are in the ratio 1 : 2 : 3, they can be represented as x, 2x, and 3x, where x is a positive real number. The cubes of these numbers are x^3, 8x^3, and 27x^3. Their sum gives an equation in x^3, which can be solved to find x, and hence the smallest number.
Step-by-Step Solution:
Step 1: Let the three numbers be x, 2x, and 3x. Step 2: Their cubes are x^3, (2x)^3 = 8x^3, and (3x)^3 = 27x^3. Step 3: Sum of cubes is given as 4500. Step 4: Form the equation: x^3 + 8x^3 + 27x^3 = 4500. Step 5: Simplify: (1 + 8 + 27)x^3 = 36x^3 = 4500. Step 6: So x^3 = 4500 / 36 = 125. Step 7: Take cube root: x = 5. Step 8: Therefore, the smallest number is x = 5.
Verification / Alternative check:
Compute the actual three numbers: 5, 10, and 15. Now check the cubes: 5^3 = 125, 10^3 = 1000, 15^3 = 3375. Sum of cubes = 125 + 1000 + 3375 = 4500, which matches the given condition exactly.
Why Other Options Are Wrong:
6: If x = 6, the cubes would sum to 36 * 6^3 = 7776, not 4500. 10: This would be the second number, not the smallest. 4: With x = 4, the sum of cubes is 36 * 64 = 2304, which is too small. 8: With x = 8, the sum of cubes is 36 * 512 = 18432, far larger than 4500.
Common Pitfalls:
Some learners mistakenly cube the ratio numbers first and then try to match them directly to 4500 without introducing a common factor x. Others may miscalculate cube values or forget to divide correctly when solving for x^3. Careful algebra and arithmetic avoid these issues.
Final Answer:
The smallest of the three numbers is 5.
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