Q4.Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by B alone?

2 days

6 days 

3 days

5 days

Cannot be determined

Let the time taken by B to finish the work be \( t_B \) days. 

- A takes twice as much time as B, so A's time is \( 2t_B \) days.
- C takes one-third of the time as A, which means C's time is \( \frac{2t_B}{3} \) days.

The work done by each individual in one day is the reciprocal of the time they take to complete the work.

- Work done by A in 1 day = \( \frac{1}{2t_B} \)
- Work done by B in 1 day = \( \frac{1}{t_B} \)
- Work done by C in 1 day = \( \frac{1}{\frac{2t_B}{3}} = \frac{3}{2t_B} \)

Since all three work together, their combined rate is:

\[
\frac{1}{2t_B} + \frac{1}{t_B} + \frac{3}{2t_B} = \frac{1 + 2 + 3}{2t_B} = \frac{6}{2t_B} = \frac{3}{t_B}
\]

It is given that they complete the entire work in 2 days. Therefore:

\[
\frac{3}{t_B} \times 2 = 1
\]

Solving for \( t_B \):

\[
\frac{6}{t_B} = 1 \quad \Rightarrow \quad t_B = 6 \text{ days}.
\]

Thus, B alone can complete the entire work in 6 days. 

Therefore, the correct answer is **6 days**.

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Do you need more details or have any questions?

Here are some related questions for practice:
1. If B alone works for 1 day, how much of the work will be completed?
2. How long will it take A and C to complete the work if B doesn't participate?
3. What is the total work done by C in 2 days?
4. If B and C work together, how long will it take them to complete the work?
5. How much work does A do in 1 day if B alone takes 6 days?

**Tip:** When solving work-rate problems, always remember that individual work rates can be added together when multiple workers are working simultaneously.

Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by B alone?

In this problem, A, B, and C are working together to complete a task in 2 days. Learn how to determine how much time B alone would take to finish the work by understanding concepts like work rate and time. This solution is ideal for students in Grades 9-10 and covers key formulas for work and time problems.

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