Answer:
To find out how many days it will take for A and B to complete the work together, we can use the concept of work rates.
Let's denote the work rate of A as \( R_A \) (the amount of work A can complete per day) and the work rate of B as \( R_B \).
Given that A completes the work in 8 days, we can express \( R_A \) as:
\[ R_A = \frac{1}{8} \text{ work per day} \]
Similarly, given that B completes the work in 24 days, we can express \( R_B \) as:
\[ R_B = \frac{1}{24} \text{ work per day} \]
When A and B work together, their combined work rate is the sum of their individual work rates:
\[ R_{\text{combined}} = R_A + R_B \]
Substituting the values of \( R_A \) and \( R_B \) into the equation:
\[ R_{\text{combined}} = \frac{1}{8} + \frac{1}{24} \]
\[ R_{\text{combined}} = \frac{3}{24} + \frac{1}{24} \]
\[ R_{\text{combined}} = \frac{4}{24} \text{ work per day} \]
\[ R_{\text{combined}} = \frac{1}{6} \text{ work per day} \]
So, when A and B work together, their combined work rate is \( \frac{1}{6} \) of the work per day.
To find out how many days it will take for them to complete the work together, we can use the formula:
\[ \text{Time} = \frac{\text{Total work}}{\text{Combined work rate}} \]
Given that the total work is 1 (the whole work), we can plug in the values:
\[ \text{Time} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]
So, it will take A and B 6 days to complete the work together.
