Of three positive numbers, the ratio of the first to the second is 5:2 and the ratio of the second to the third is 5:4. If the product of the first and the third numbers is 1,800, what is the sum of the three numbers?

Introduction / Context:
This question links two ratio relationships and a given product of two numbers to find the sum of all three numbers. It is a nice example of how proportional reasoning can be combined with an additional numerical condition to determine exact values rather than just relative sizes. Such problems are common in competitive exams to test understanding of ratios and algebra.

Given Data / Assumptions:

Three positive numbers are involved.
First : Second = 5 : 2.
Second : Third = 5 : 4.
The product of the first and the third numbers is 1,800.
We are asked for the sum of all three numbers.

Concept / Approach:
We first express the three numbers in terms of a common parameter by combining the two given ratios. Once we obtain a consistent combined ratio for first, second and third, we use the product of the first and third numbers to determine the value of the scaling parameter. After that, we compute each number and finally sum them. This approach avoids guesswork and uses systematic algebra.

Step-by-Step Solution:
Let the three numbers be A, B and C. From the first ratio, A : B = 5 : 2, so let A = 5x and B = 2x. From the second ratio, B : C = 5 : 4, so let B = 5y and C = 4y. We need the two expressions for B to be equal, so 2x = 5y. Set 2x = 5y = some common value, for convenience take 2x = 5y = 10k, giving x = 5k and y = 2k. Then A = 5x = 5 * 5k = 25k, B = 2x = 10k, and C = 4y = 4 * 2k = 8k. Given that A * C = 1,800, so 25k * 8k = 1,800. Thus 200k^2 = 1,800, which implies k^2 = 1,800 / 200 = 9. Therefore k = 3 (positive because numbers are positive). Now A = 25k = 75, B = 10k = 30, C = 8k = 24. Sum of the three numbers = 75 + 30 + 24 = 129.

Verification / Alternative check:
Check both ratios and the product. A:B = 75:30 simplifies to 5:2, and B:C = 30:24 simplifies to 5:4, satisfying the ratio conditions. The product A * C = 75 * 24 = 1,800, matching the given data. Therefore the sum 129 is consistent with all parts of the question.

Why Other Options Are Wrong:
Option 43 is far too small and would correspond to much smaller numbers that cannot have a product of 1,800 when multiplying the first and third.
Option 133 would require slightly different scaling values that would break either the ratios or the product condition.
Option 119 also does not arise from any integer scaling factor k that simultaneously satisfies both ratios and the given product of 1,800.

Common Pitfalls:
Some students try to add the ratios directly without first aligning the common term B, which leads to inconsistent numbers.
Another error is to forget that the product condition A * C = 1,800 must hold exactly and treat it as approximate, leading to rounding mistakes.
Learners also sometimes take k^2 = 9 but then accidentally use k = 9 instead of k = 3, which makes the numbers much larger and inconsistent with the product condition.

Final Answer:
The sum of the three numbers is 129.