Question

Fifty is divided into two parts such that the sum of their reciprocals is \( \frac{1}{12} \). Find the two parts.

• A. \( 25, 35 \)
• B. \( 30, 20 \)
• C. \( 15, 35 \)
• D. \( 20, 20 \)
• E.

Hint:

For problems involving the sum of reciprocals, form a quadratic equation and solve for the unknowns.

Answer 1

The Correct Option is B

Let the two parts be \( x \) and \( 50 - x \). 

Step 1: The sum of their reciprocals is given by: \[ \frac{1}{x} + \frac{1}{50 - x} = \frac{1}{12} \] 

Step 2: Multiply both sides by \( 12x(50 - x) \): \[ 12(50 - x) + 12x = x(50 - x) \] 

Step 3: Simplifying the equation: \[ 600 - 12x + 12x = 50x - x^2 \] \[ 600 = 50x - x^2 \] 

Step 4: Rearranging into a quadratic equation: \[ x^2 - 50x + 600 = 0 \] 

Step 5: Solving the quadratic equation using the quadratic formula: \[ x = \frac{50 \pm \sqrt{50^2 - 4(1)(600)}}{2} \] \[ x = \frac{50 \pm \sqrt{2500 - 2400}}{2} \] \[ x = \frac{50 \pm 10}{2} \] 

Step 6: The solutions are: \[ x = \frac{50 + 10}{2} = 30 \quad \text{or} \quad x = \frac{50 - 10}{2} = 20 \] Thus, the two parts are 30 and 20.