A and B complete a work in 6 days. A alone can do it in 10 days. If both together can do the work in how many days?a. 6 daysb. 4 daysc. 5 daysd. 3.75 days

1. Break Down the Problem

• Let the total work be represented in terms of work done per day by A and B.
• A completes the work in 10 days, hence A's rate is $\frac{1}{10}$ of the work per day.
• When A and B work together, they complete the work in 6 days, meaning their combined rate is $\frac{1}{6}$ of the work per day.

2. Relevant Concepts

• The equation for combined rates where A's rate is added to B's rate: $\text{Rate of A} + \text{Rate of B} = \text{Combined Rate}$
• From the problem, we can express this mathematically as: $\frac{1}{10} + \text{Rate of B} = \frac{1}{6}$

3. Analysis and Detail

• To find the rate of B, rearranging gives: $\text{Rate of B} = \frac{1}{6} - \frac{1}{10}$
• To perform this subtraction, we need a common denominator: $\text{Common Denominator} = 30$ $\frac{1}{6} = \frac{5}{30}, \quad \frac{1}{10} = \frac{3}{30}$ $\text{Rate of B} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15}$

4. Verify and Summarize

• Now, we have A's rate as $\frac{1}{10}$ and B's rate as $\frac{1}{15}$.
• Their combined rate: $\text{Combined Rate} = \frac{1}{10} + \frac{1}{15}$
• Finding a common denominator (30), we get: $\frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$
• This confirms our earlier finding that together they can finish the work in 6 days.

Final Answer

The answer is $a. 6$ days.