Sum of two numbers is 40 and their product is 375. What will be the sum of their reciprocals?

Options

A. `(8)/(75)`
B. `(1)/(40)`
C. `(75)/(8)`
D. `(75)/(4)`

Answer

To solve the problem, we need to find the sum of the reciprocals of two numbers, given that their sum is 40 and their product is 375.

Let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations
From the problem, we have:
1. \( x + y = 40 \) (Equation 1)
2. \( xy = 375 \) (Equation 2)

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 40 - x \]

### Step 3: Substitute into the product equation
Now, substitute \( y \) in Equation 2:
\[ x(40 - x) = 375 \]

### Step 4: Rearrange the equation
Expanding the equation gives:
\[ 40x - x^2 = 375 \]
Rearranging it, we get:
\[ x^2 - 40x + 375 = 0 \]

### Step 5: Solve the quadratic equation
Now we will use the quadratic formula to find the values of \( x \):
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -40 \), and \( c = 375 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-40)^2 - 4 \cdot 1 \cdot 375 = 1600 - 1500 = 100 \]

Now substituting into the quadratic formula:
\[ x = \frac{40 \pm \sqrt{100}}{2 \cdot 1} \]
\[ x = \frac{40 \pm 10}{2} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{50}{2} = 25 \)
2. \( x = \frac{30}{2} = 15 \)

### Step 6: Find the values of \( y \)
Using \( y = 40 - x \):
1. If \( x = 25 \), then \( y = 40 - 25 = 15 \)
2. If \( x = 15 \), then \( y = 40 - 15 = 25 \)

So the two numbers are \( 25 \) and \( 15 \).

### Step 7: Find the sum of the reciprocals
The sum of the reciprocals of \( x \) and \( y \) is given by:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{25} + \frac{1}{15} \]

To add these fractions, we need a common denominator:
The least common multiple of \( 25 \) and \( 15 \) is \( 75 \).

Converting the fractions:
\[ \frac{1}{25} = \frac{3}{75} \]
\[ \frac{1}{15} = \frac{5}{75} \]

Now adding them:
\[ \frac{3}{75} + \frac{5}{75} = \frac{8}{75} \]

### Final Answer
The sum of the reciprocals of the two numbers is:
\[ \frac{8}{75} \]

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