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

Correct option is B

Given:

- First term = x

- Second term = x + 100

- Third proportional = 405

Formula Used:

- If a, b are two numbers, then third proportional = $$\frac{b² }{ a}$$​​

Solution:

According to the formula:

Third proportional = $$\frac{(x + 100)² }{ x }= 405$$​​

=> (x + 100)² = 405x

=> x² + 200x + 10000 = 405x

=> x² − 205x + 10000 = 0

Solve the quadratic equation:

x² − 205x + 10000 = 0

​$$x = \frac{-(-205) \pm \sqrt{205^2 - 4 \cdot 1 \cdot 10000}}{2 \cdot 1} \\\ \\= \frac{205 \pm \sqrt{42025 - 40000}}{2} \\\ \\= \frac{205 \pm \sqrt{2025}}{2} \\\ \\= \frac{205 \pm 45}{2} \\\ \\x = \frac{205 + 45}{2} = \frac{250}{2} = 125 \quad (\text{valid since } x > 100) \\\ \\\text{or } x = \frac{205 - 45}{2} = \frac{160}{2} = 80 \quad (\text{rejected since } x < 100)$$​​