Explanation:

- You’re given ratios and a product for three positive numbers.

- Let’s say the first number is \(5x\), and the second is \(2x\) since the ratio is 5:2.

- The ratio of the second number to the third is \(5:4\), thus the third number is \(\frac{4}{5} \times 2x = \frac{8}{5}x\).

- The product of the first and third numbers is \(5x \times \frac{8}{5}x = \frac{40}{5}x^2\).

- Given that product is 1800, solve: \(8x^2 = 1800\).

- Divide both sides by 8: \(x^2 = 225\).

- \(x = 15\).

- Calculate each number:

- First: \(5x = 75\)

- Second: \(2x = 30\)

- Third: \(\frac{8}{5}x = 24\)

- Sum of numbers: \(75 + 30 + 24 = 129\).

- Correct Answer: Option 4, 129