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 their sum and product. Let's break it down step by step.

### Step 1: Define the variables
Let the two numbers be \( A \) and \( B \).

### Step 2: Write down the equations
According to the problem, we have:
1. \( A + B = 40 \) (the sum of the numbers)
2. \( A \times B = 375 \) (the product of the numbers)

### Step 3: Find the sum of the reciprocals
The sum of the reciprocals of \( A \) and \( B \) can be expressed as:
\[ \frac{1}{A} + \frac{1}{B} = \frac{B + A}{A \times B} \]
Using the equations from Step 2, we can substitute:
\[ \frac{1}{A} + \frac{1}{B} = \frac{A + B}{A \times B} = \frac{40}{375} \]

### Step 4: Simplify the fraction
Now, we simplify \( \frac{40}{375} \):
1. Find the greatest common divisor (GCD) of 40 and 375. The GCD is 5.
2. Divide both the numerator and the denominator by their GCD:
\[ \frac{40 \div 5}{375 \div 5} = \frac{8}{75} \]

### Step 5: Conclusion
Thus, the sum of the reciprocals of the two numbers is:
\[ \frac{8}{75} \]

### Final Answer
The sum of the reciprocals of the two numbers is \( \frac{8}{75} \).

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