Math Problem Statement

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

Solution

Let the time taken by B to finish the work be tBt_B days.

  • A takes twice as much time as B, so A's time is 2tB2t_B days.
  • C takes one-third of the time as A, which means C's time is 2tB3\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 = 12tB\frac{1}{2t_B}
  • Work done by B in 1 day = 1tB\frac{1}{t_B}
  • Work done by C in 1 day = 12tB3=32tB\frac{1}{\frac{2t_B}{3}} = \frac{3}{2t_B}

Since all three work together, their combined rate is:

12tB+1tB+32tB=1+2+32tB=62tB=3tB\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:

3tB×2=1\frac{3}{t_B} \times 2 = 1

Solving for tBt_B:

6tB=1tB=6 days.\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.


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.

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Math Problem Analysis

Mathematical Concepts

Work and Time Problems
Ratio and Proportion

Formulas

Work done = Rate × Time
Combined work rate = Sum of individual work rates

Theorems

Work-rate theorem: The combined work rate is the sum of individual rates.

Suitable Grade Level

Grades 9-10