Question

Amanda, Beth, and Sam can write a program in 3 days. Amanda can write the program in 8 days, and Beth can do it in 10 days. How many days will it take Sam to write the program?

Answer 1

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:
\[
\boxed{\frac{120}{13}} \text{ days} \quad \text{(approximately 9.23 days)}
\]