A can complete a piece of work in 10 days, B in 15 days and C in 20 days. A and C worked together for two days and then A was replaced by B. In how many days, altogether, was the work completed ?

Options

A. 12 days
B. 10 days
C. 6 days
D. 8 days

Answer

To solve the problem step by step, let's break down the work done by A, B, and C and how the work progresses over the days.

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 10 days, so in one day, A does:
\[ \text{Work done by A in one day} = \frac{1}{10} \text{ of the work} \]
- B can complete the work in 15 days, so in one day, B does:
\[ \text{Work done by B in one day} = \frac{1}{15} \text{ of the work} \]
- C can complete the work in 20 days, so in one day, C does:
\[ \text{Work done by C in one day} = \frac{1}{20} \text{ of the work} \]

### Step 2: Calculate the total work done by A and C together in one day.
- The combined work done by A and C in one day is:
\[ \text{Work done by A and C in one day} = \frac{1}{10} + \frac{1}{20} \]
To add these fractions, we find a common denominator (which is 20):
\[ = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \]

### Step 3: Calculate the work done by A and C in 2 days.
- In 2 days, A and C together will complete:
\[ \text{Work done in 2 days} = 2 \times \frac{3}{20} = \frac{6}{20} = \frac{3}{10} \]

### Step 4: Determine the remaining work after A and C have worked for 2 days.
- The total work is considered as 1 (or 100% of the work). Therefore, the remaining work after A and C have completed \(\frac{3}{10}\) of the work is:
\[ \text{Remaining work} = 1 - \frac{3}{10} = \frac{7}{10} \]

### Step 5: Calculate the work done by B and C together in one day.
- Now, A is replaced by B. The combined work done by B and C in one day is:
\[ \text{Work done by B and C in one day} = \frac{1}{15} + \frac{1}{20} \]
To add these fractions, we find a common denominator (which is 60):
\[ = \frac{4}{60} + \frac{3}{60} = \frac{7}{60} \]

### Step 6: Determine how many days B and C will take to complete the remaining work.
- The remaining work is \(\frac{7}{10}\). To find out how many days it will take for B and C to complete this work, we set up the equation:
\[ \text{Days} = \frac{\text{Remaining work}}{\text{Work done by B and C in one day}} = \frac{\frac{7}{10}}{\frac{7}{60}} = \frac{7}{10} \times \frac{60}{7} = 6 \text{ days} \]

### Step 7: Calculate the total time taken to complete the work.
- The total time taken to complete the work is the time A and C worked together plus the time B and C worked together:
\[ \text{Total time} = 2 \text{ days} + 6 \text{ days} = 8 \text{ days} \]

### Final Answer:
The total time taken to complete the work is **8 days**.

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