Let the amount of work required to complete the job be represented as $W$.

Step 1: Defining efficiency and time for A, B, and C

Efficiency of B:

Let B's rate of work (efficiency) be $r_B$ units of work per day. Thus, B takes $\frac{W}{r_B}$ days to complete the job.

Efficiency of A:

A is 20% more efficient than B, meaning A's rate of work is $r_A = 1.2r_B$.

Let the time taken by B to complete the job be $t_B$ days. From the problem, we know that A takes 40 days less than B to complete the job: $t_A = t_B - 40$ Since the time to complete the job is the total work divided by the rate of work, we know: $t_A = \frac{W}{r_A} = \frac{W}{1.2r_B}$ $t_B = \frac{W}{r_B}$ Thus, from the given information, we have: $\frac{W}{1.2r_B} = \frac{W}{r_B} - 40$ Multiplying through by $1.2r_B$ to eliminate the denominators: $W = 1.2W - 40 \times 1.2r_B$ $0.2W = 40 \times 1.2r_B$ $r_B = \frac{0.2W}{40 \times 1.2} = \frac{W}{240}$ Thus, B's rate of work is $r_B = \frac{W}{240}$.

Efficiency of C:

B is 50% more efficient than C, meaning B’s rate of work is 1.5 times that of C. Hence, C's rate of work is: $r_C = \frac{r_B}{1.5} = \frac{W}{240 \times 1.5} = \frac{W}{360}$

Step 2: Amount of work completed by A, B, and C

Work done by A in 50 days:

A works for 50 days, so the work done by A is: $\text{Work by A} = r_A \times 50 = 1.2r_B \times 50 = 1.2 \times \frac{W}{240} \times 50 = \frac{1.2W \times 50}{240} = \frac{W}{4}$ Thus, A completes $\frac{W}{4}$ of the total work.

Work done by B in 20 days:

B works for 20 days, so the work done by B is: $\text{Work by B} = r_B \times 20 = \frac{W}{240} \times 20 = \frac{W}{12}$ Thus, B completes $\frac{W}{12}$ of the total work.

Remaining work for C:

The total work done by A and B is: $\frac{W}{4} + \frac{W}{12} = \frac{3W}{12} + \frac{W}{12} = \frac{4W}{12} = \frac{W}{3}$ Thus, the remaining work for C is: $W - \frac{W}{3} = \frac{2W}{3}$

Step 3: Time taken by C to complete the remaining work

The time taken by C to complete $\frac{2W}{3}$ of the total work is: $\text{Time by C} = \frac{\frac{2W}{3}}{r_C} = \frac{\frac{2W}{3}}{\frac{W}{360}} = \frac{2W}{3} \times \frac{360}{W} = \frac{2 \times 360}{3} = 240 \text{ days}$

Final Answer:

C takes 240 days to complete the remaining work.

Let me know if you'd like more details or have any questions.

Here are 5 related questions for further practice:

1. If A worked for 40 days instead of 50, how many days would C take to complete the remaining work?
2. If C were 25% more efficient, how many days would C take to complete the work left after A and B?
3. How long would it take A to complete the entire job on their own?
4. If B worked for 30 days instead of 20, how much work would be left for C to complete?
5. If C worked for the first 60 days, how much work would be left for A and B to complete?

Tip: Efficiency comparisons are often relative, so always express one worker's efficiency in terms of another to simplify the calculations.