A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 6 days. B can do the work alone in?

Let's define the time each person takes to complete the work individually:

- Let \( B \) take \( x \) days to complete the work.
- Since \( A \) takes twice as much time as \( B \), \( A \) will take \( 2x \) days.
- \( A \) also takes thrice as much time as \( C \), so \( C \) will take \(\frac{2x}{3} \) days.

### Step-by-Step Solution

1. **Determine the work rates of A, B, and C:**

   The rate of work (work per day) of each person is the reciprocal of the time they take to complete the work.

   - Rate of \( A = \frac{1}{2x} \) (work per day)
   - Rate of \( B = \frac{1}{x} \) (work per day)
   - Rate of \( C = \frac{3}{2x} \) (work per day)

2. **Calculate their combined work rate:**

   When they work together, their combined rate is the sum of their individual rates:

   \[
   \text{Combined rate} = \frac{1}{2x} + \frac{1}{x} + \frac{3}{2x}
   \]

   Combine the fractions:

   \[
   \text{Combined rate} = \frac{1 + 2 + 3}{2x} = \frac{6}{2x} = \frac{3}{x} \, \text{(work per day)}
   \]

3. **Use the information that they finish the work in 6 days:**

   If they complete the work together in 6 days, then:

   \[
   6 \times \text{Combined rate} = 1 \, \text{(whole work)}
   \]

   Substituting the combined rate:

   \[
   6 \times \frac{3}{x} = 1
   \]

   Solve for \( x \):

   \[
   \frac{18}{x} = 1 \implies x = 18
   \]

4. **Find the time taken by B to do the work alone:**

   Since \( x = 18 \), \( B \) takes \( 18 \) days to complete the work alone.

### Conclusion

\( B \) can complete the work alone in **18 days**.

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How long would it take for \( A \) to finish the work alone?
2. What is the time required for \( C \) to finish the work alone?
3. If all three work together but \( A \) works only half the time, how long will it take to finish the work?
4. What fraction of the work does each person complete if they all work together for 3 days?
5. How would the solution change if \( A \) took four times as long as \( C \) to finish the work?

**Tip:** When solving work problems, always set up relationships between the rates of work to make solving easier.

Learn how to calculate the time B takes to complete work alone in a time and work problem where A takes twice as much time as B and thrice as much time as C. Understand the combined work rate concept and its application. Suitable for Grades 7-9 students.

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