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

Answer 1

The Correct Option is B

Step 1: Define the parts 

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

Step 2: Use the given condition

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

Step 3: Multiply both sides by \( 12x(50 - x) \) to eliminate the fractions

\[ 12(50 - x) + 12x = x(50 - x) \]

Step 4: Simplify the equation

\[ 600 - 12x + 12x = 50x - x^2 \] \[ 600 = 50x - x^2 \]

Step 5: Rearrange into a quadratic equation

\[ x^2 - 50x + 600 = 0 \]

Step 6: Solve for \( x \)

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

Step 7: Find the possible values of \( x \)

\[ x = \frac{50 + 10}{2} = \frac{60}{2} = 30 \] \[ x = \frac{50 - 10}{2} = \frac{40}{2} = 20 \]

Thus, the two parts are 30 and 20.