Divide 50 into two parts so that the sum of their reciprocals is `(1)/(12)`

Options

A. 35,15
B. 20,30
C. 24,36
D. 28,22

Answer

To solve the problem of dividing 50 into two parts such that the sum of their reciprocals is \( \frac{1}{12} \), we can follow these steps:

### Step 1: Define the variables
Let the two parts be \( x \) and \( y \). According to the problem, we know that:
\[ x + y = 50 \]

### Step 2: Express one variable in terms of the other
From the equation \( x + y = 50 \), we can express \( y \) in terms of \( x \):
\[ y = 50 - x \]

### Step 3: Set up the equation for the sum of reciprocals
The problem states that the sum of the reciprocals of \( x \) and \( y \) is \( \frac{1}{12} \):
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \]

### Step 4: Substitute \( y \) in the equation
Substituting \( y = 50 - x \) into the equation gives:
\[ \frac{1}{x} + \frac{1}{50 - x} = \frac{1}{12} \]

### Step 5: Find a common denominator
To combine the fractions on the left side, we find a common denominator:
\[ \frac{(50 - x) + x}{x(50 - x)} = \frac{1}{12} \]
This simplifies to:
\[ \frac{50}{x(50 - x)} = \frac{1}{12} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 50 \cdot 12 = x(50 - x) \]
This simplifies to:
\[ 600 = 50x - x^2 \]

### Step 7: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:
\[ x^2 - 50x + 600 = 0 \]

### Step 8: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -50 \), and \( c = 600 \):
\[ x = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 1 \cdot 600}}{2 \cdot 1} \]
\[ x = \frac{50 \pm \sqrt{2500 - 2400}}{2} \]
\[ x = \frac{50 \pm \sqrt{100}}{2} \]
\[ x = \frac{50 \pm 10}{2} \]

### Step 9: Calculate the possible values for \( x \)
Calculating the two possible values for \( x \):
1. \( x = \frac{60}{2} = 30 \)
2. \( x = \frac{40}{2} = 20 \)

### Step 10: Find the corresponding values for \( y \)
Using \( y = 50 - x \):
1. If \( x = 30 \), then \( y = 50 - 30 = 20 \)
2. If \( x = 20 \), then \( y = 50 - 20 = 30 \)

Thus, the two parts are \( 30 \) and \( 20 \).

### Final Answer
The two parts are \( 30 \) and \( 20 \).

---