Divide 15 into two parts such that the sum of their reciprocals is `(3)/(10).

Answer

To solve the problem of dividing 15 into two parts such that the sum of their reciprocals is \( \frac{3}{10} \), we can follow these steps:

### Step 1: Define the Variables
Let the two parts be \( x \) and \( y \). According to the problem, we have:
\[ x + y = 15 \]

### Step 2: Write the Equation for Reciprocals
The sum of their reciprocals is given as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{3}{10} \]

### Step 3: Find a Common Denominator
To combine the fractions on the left side, we can find a common denominator:
\[ \frac{y + x}{xy} = \frac{3}{10} \]
Since \( x + y = 15 \), we can substitute this into the equation:
\[ \frac{15}{xy} = \frac{3}{10} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 15 \cdot 10 = 3 \cdot xy \]
This simplifies to:
\[ 150 = 3xy \]

### Step 5: Solve for \( xy \)
Now, divide both sides by 3:
\[ xy = \frac{150}{3} = 50 \]

### Step 6: Set Up the Quadratic Equation
We now have two equations:
1. \( x + y = 15 \)
2. \( xy = 50 \)

We can express \( y \) in terms of \( x \):
\[ y = 15 - x \]
Substituting this into the second equation gives:
\[ x(15 - x) = 50 \]
Expanding this, we get:
\[ 15x - x^2 = 50 \]

### Step 7: Rearrange into Standard Form
Rearranging the equation to standard quadratic form:
\[ x^2 - 15x + 50 = 0 \]

### Step 8: Factor the Quadratic
Next, we need to factor the quadratic equation. We look for two numbers that multiply to 50 and add up to -15. The factors are:
\[ (x - 10)(x - 5) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 10 = 0 \quad \text{or} \quad x - 5 = 0 \]
Thus,
\[ x = 10 \quad \text{or} \quad x = 5 \]

### Step 10: Find the Corresponding \( y \) Values
Using \( y = 15 - x \):
- If \( x = 10 \), then \( y = 15 - 10 = 5 \).
- If \( x = 5 \), then \( y = 15 - 5 = 10 \).

### Conclusion
The two parts are \( 10 \) and \( 5 \).