Question

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

• A. 225
• B. 125
• C. 115
• D. 180

Answer 1

Answer:

B. 125

The third proportional to two numbers (a) and (b) is given by:
$\frac{a}{b} = \frac{b}{c}$
$\Rightarrow c = \frac{b^2}{a}$

Here,
$(a = x), (b = x + 100)$, and third proportional $(c = 405)$

So,
$405 = \frac{(x + 100)^2}{x}$

Multiply both sides by (x):
$405x = (x + 100)^2$
$405x = x^2 + 200x + 10000$

Bring all terms to one side:
$x^2 - 205x + 10000 = 0$

Solve the quadratic:

$x = \frac{205 \pm \sqrt{205^2 - 4 \cdot 10000}}{2}$

$= \frac{205 \pm \sqrt{42025 - 40000}}{2}$
$= \frac{205 \pm \sqrt{2025}}{2}$
$= \frac{205 \pm 45}{2}$

So,
$x = \frac{250}{2} = 125 \quad \text{or} \quad x = \frac{160}{2} = 80$

Given (x > 100),

Final Answer:

x = 125