Number 60 is divided into two parts such that the sum of their reciprocals is `(3)/(40).` Find the two parts.

Options

A. `40,20`
B. `30,30`
C. `10,50`
D. `5,15`

Answer

To solve the problem of dividing the number 60 into two parts such that the sum of their reciprocals is \( \frac{3}{40} \), we can follow these steps:

### Step 1: Define the Parts
Let the first part be \( x \). Then, the second part will be:
\[ 60 - x \]

### Step 2: Write the Equation for Reciprocals
According to the problem, the sum of the reciprocals of these two parts is given by:
\[ \frac{1}{x} + \frac{1}{60 - x} = \frac{3}{40} \]

### Step 3: Find a Common Denominator
To combine the fractions on the left side, we need a common denominator:
\[ \frac{(60 - x) + x}{x(60 - x)} = \frac{3}{40} \]
This simplifies to:
\[ \frac{60}{x(60 - x)} = \frac{3}{40} \]

### Step 4: Cross Multiply
Now, we can cross-multiply to eliminate the fractions:
\[ 60 \cdot 40 = 3 \cdot x(60 - x) \]
This simplifies to:
\[ 2400 = 3x(60 - x) \]

### Step 5: Expand and Rearrange
Expanding the right side gives:
\[ 2400 = 180x - 3x^2 \]
Rearranging this equation leads to:
\[ 3x^2 - 180x + 2400 = 0 \]

### Step 6: Simplify the Quadratic Equation
Dividing the entire equation by 3 simplifies it:
\[ x^2 - 60x + 800 = 0 \]

### Step 7: Factor the Quadratic Equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to 800 and add up to -60. These numbers are -40 and -20. Thus, we can factor the equation as:
\[ (x - 40)(x - 20) = 0 \]

### Step 8: Solve for x
Setting each factor to zero gives us:
\[ x - 40 = 0 \quad \Rightarrow \quad x = 40 \]
\[ x - 20 = 0 \quad \Rightarrow \quad x = 20 \]

### Step 9: Find the Two Parts
Now we can find the two parts:
1. If \( x = 40 \), then the second part is \( 60 - 40 = 20 \).
2. If \( x = 20 \), then the second part is \( 60 - 20 = 40 \).

Thus, the two parts are \( 40 \) and \( 20 \).

### Final Answer
The two parts are \( 40 \) and \( 20 \).

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