The third proportional to x and x + 100 is 405, find the value of x (where x > 100).

Given:

First proportional = x

Second proportional = x + 100

Third proportional = 405

Condition: x > 100

Formula Used:

If a, b, and c are in continued proportion, then b² = ac.

Here, a = x, b = x + 100, and c = 405.

Calculation:

Using the formula for continued proportion:

(x + 100)² = x × 405

⇒ x² + 2 × x × 100 + 100² = 405x

⇒ x² + 200x + 10000 = 405x

⇒ x² + 200x - 405x + 10000 = 0

⇒ x² - 205x + 10000 = 0

Solve the quadratic equation using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

Here, a = 1, b = -205, c = 10000.

Discriminant (D) = b² - 4ac = (-205)² - 4 × 1 × 10000

D = 42025 - 40000

D = 2025

√D = √2025 = 45

Now, find the values of x:

x₁ = [-(-205) + 45] / (2 × 1) = (205 + 45) / 2 = 250 / 2 = 125

x₂ = [-(-205) - 45] / (2 × 1) = (205 - 45) / 2 = 160 / 2 = 80

We are given that x > 100.

Therefore, the value of x is 125.

∴ The value of x is 125.