- **Identifying the Consecutive Numbers**
First, let's assume the two consecutive numbers as \(x\) and \(x+1\). Therefore, the reciprocals of these numbers will be \(\frac{1}{x}\) and \(\frac{1}{x+1}\) respectively.
- **Formulating the Equation**
Given that the difference of the reciprocals is \( \frac{1}{x} - \frac{1}{x+1} = \frac{1}{40} \), we can simplify this equation by finding a common denominator and solving for \(x\).
- **Solving the Equation**
\( \frac{x+1 - x}{x(x+1)} = \frac{1}{40} \)
\( \frac{1}{x(x+1)} = \frac{1}{40} \)
By cross multiplying, we get:
\( 40 = x(x+1) \)
- **Finding the Numbers**
Now, we need to solve the quadratic equation \(x^2 + x - 40 = 0\) to find the values of \(x\). The solutions to this equation will give us the two consecutive numbers that satisfy the given condition.
- **Solution**
By solving the quadratic equation, we get two possible values for \(x\): \(x = -8\) or \(x = 5\). Since we are dealing with consecutive numbers, we will consider the positive value \(x = 5\).
Therefore, the two consecutive numbers are 5 and 6, as their reciprocals \(\frac{1}{5}\) and \(\frac{1}{6}\) have a difference of \(\frac{1}{40}\).