A is thrice as good a workman as B and therefore is able to finish a job in 80 days less than B. If A and B complete `(5)/(8)`th of the job and then A is replaced by C, the remaining job is done by B and C in 15 days. If A and B complete `(5)/(12)`th of the job and then B is replaced by D the remaining job is done by A and D in 10 days.
How long will it take to complete the job, if A,B,C and D working together?

Options

A. 12 days
B. 10 days
C. 15 days
D. 18 days

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, C, and D, and then calculate how long it will take for all of them to complete the job together.

### Step 1: Determine the efficiencies of A and B
Given that A is thrice as good a workman as B, we can denote:
- Efficiency of B = 1 unit of work per day
- Efficiency of A = 3 units of work per day

Let the time taken by B to complete the job be \( x \) days. Since A finishes the job in 80 days less than B, we have:
\[ \text{Time taken by A} = x - 80 \]

Using the formula for work:
\[ \text{Total Work} = \text{Efficiency} \times \text{Time} \]
For A:
\[ 3(x - 80) = \text{Total Work} \]
For B:
\[ 1 \times x = \text{Total Work} \]

Setting these equal gives:
\[ 3(x - 80) = x \]
Expanding and simplifying:
\[ 3x - 240 = x \]
\[ 2x = 240 \]
\[ x = 120 \]

So, B takes 120 days to complete the job, and A takes:
\[ 120 - 80 = 40 \text{ days} \]

### Step 2: Calculate the total work
Using B's time:
\[ \text{Total Work} = 1 \times 120 = 120 \text{ units} \]

### Step 3: Work done by A and B together
A and B together complete \( \frac{5}{8} \) of the job:
\[ \text{Work done} = \frac{5}{8} \times 120 = 75 \text{ units} \]

### Step 4: Remaining work
Remaining work after A and B:
\[ 120 - 75 = 45 \text{ units} \]

### Step 5: Work done by B and C
B and C complete the remaining 45 units in 15 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of B and C} = \frac{45}{15} = 3 \text{ units per day} \]

Since B's efficiency is 1 unit per day, we can find C's efficiency:
\[ 1 + C = 3 \]
\[ C = 2 \text{ units per day} \]

### Step 6: Work done by A and B together again
Next, A and B complete \( \frac{5}{12} \) of the job:
\[ \text{Work done} = \frac{5}{12} \times 120 = 50 \text{ units} \]

### Step 7: Remaining work
Remaining work after A and B:
\[ 120 - 50 = 70 \text{ units} \]

### Step 8: Work done by A and D
A and D complete the remaining 70 units in 10 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A and D} = \frac{70}{10} = 7 \text{ units per day} \]

Since A's efficiency is 3 units per day, we can find D's efficiency:
\[ 3 + D = 7 \]
\[ D = 4 \text{ units per day} \]

### Step 9: Total efficiency of A, B, C, and D together
Now, we can find the total efficiency when A, B, C, and D work together:
\[ \text{Total Efficiency} = A + B + C + D = 3 + 1 + 2 + 4 = 10 \text{ units per day} \]

### Step 10: Time taken to complete the job together
Finally, to find the time taken to complete the entire job when A, B, C, and D work together:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{120}{10} = 12 \text{ days} \]

### Final Answer
Thus, it will take **12 days** to complete the job if A, B, C, and D work together.

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