The sum of two numbers is 10. Their product is 20. Find the sum of the reciprocals of the two numbers.

Options

A. 1
B. `3/5`
C. `1/2`
D. `11/6`

Answer

To solve the problem step by step, we start with the information given:

1. **Let the two numbers be x and y.**
- We know that:
- \( x + y = 10 \) (Equation 1)
- \( x \cdot y = 20 \) (Equation 2)

2. **We need to find the sum of the reciprocals of the two numbers, which is given by:**
\[ \frac{1}{x} + \frac{1}{y} \]

3. **We can rewrite the sum of the reciprocals using a common denominator:**
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} \]
- From Equation 1, we know \( x + y = 10 \).
- From Equation 2, we know \( x \cdot y = 20 \).

4. **Substituting the values from Equations 1 and 2 into the expression for the sum of the reciprocals:**
\[ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} = \frac{10}{20} \]

5. **Now, simplify the fraction:**
\[ \frac{10}{20} = \frac{1}{2} \]

6. **Thus, the sum of the reciprocals of the two numbers is:**
\[ \frac{1}{2} \]

### Summary of Steps:
1. Define the two numbers and write their sum and product as equations.
2. Write the sum of the reciprocals using a common denominator.
3. Substitute the known values into the expression.
4. Simplify the resulting fraction.