To determine how long it will take Sam to write the program, we start by figuring out the work rate of each individual.
Let:
- \( A \) be the rate at which Amanda works,
- \( B \) be the rate at which Beth works,
- \( S \) be the rate at which Sam works.
From the problem statement, we know the following:
- Amanda can complete the program in 8 days, so her work rate is: \[ A = \frac{1}{8} \text{ programs per day} \]
- Beth can complete the program in 10 days, so her work rate is: \[ B = \frac{1}{10} \text{ programs per day} \]
- Together, Amanda, Beth, and Sam can complete the program in 3 days, so their combined work rate is: \[ A + B + S = \frac{1}{3} \text{ programs per day} \]
Now we can substitute the values of \( A \) and \( B \) into the equation: \[ \frac{1}{8} + \frac{1}{10} + S = \frac{1}{3} \]
Next, we need to find a common denominator to simplify \( \frac{1}{8} + \frac{1}{10} \). The least common multiple of 8 and 10 is 40. We can convert each fraction as follows: \[ \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40} \] Thus, we have: \[ \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \]
Now, substituting this back into our equation gives: \[ \frac{9}{40} + S = \frac{1}{3} \]
Next, we convert \( \frac{1}{3} \) to a fraction with a denominator of 40: \[ \frac{1}{3} = \frac{40}{120} = \frac{40}{120} \cdot \frac{40}{40} = \frac{40}{120} \]
Next, we need to find a common denominator for \( \frac{1}{3} \) and \( \frac{9}{40} \). The least common multiple of 3 and 40 is 120. We convert: \[ \frac{1}{3} = \frac{40}{120} \quad \text{and} \quad \frac{9}{40} = \frac{27}{120} \]
Now we substitute: \[ \frac{27}{120} + S = \frac{40}{120} \]
This implies: \[ S = \frac{40}{120} - \frac{27}{120} = \frac{13}{120} \]
So Sam's work rate is \( S = \frac{13}{120} \) programs per day.
To find out how many days it will take Sam to write the program alone, we compute the reciprocal of his rate: \[ \text{Days for Sam} = \frac{1}{S} = \frac{1}{\frac{13}{120}} = \frac{120}{13} \approx 9.23 \text{ days} \]
Thus, Sam will take approximately \( \frac{120}{13} \) or about 9.23 days to complete the program alone.
The final answer is: \[ \boxed{\frac{120}{13}} \text{ days} \quad \text{(approximately 9.23 days)} \]