A can do a piece of work in 2 hours, B can do thrice the work in 8 hours and C can do the same work as A in 8 hours. If all of them work together, how long it would take them to complete the work :

Options

A. 1 hour
B. 2 hours
C. 3 hours
D. 4 hours

Answer

To solve the problem step by step, let's first summarize the information given:

1. A can complete the work in 2 hours.
2. B can complete thrice the work in 8 hours, which means B can complete 1 work in \( \frac{8}{3} \) hours.
3. C can do the same work as A in 8 hours.

Now, let's calculate how much work each person can do in one hour:

### Step 1: Calculate the work done by A in one hour
A completes the work in 2 hours, so in one hour, A can do:
\[ \text{Work done by A in 1 hour} = \frac{1 \text{ work}}{2 \text{ hours}} = 0.5 \text{ work/hour} \]

### Step 2: Calculate the work done by B in one hour
B can do thrice the work in 8 hours, which means he can do 1 work in \( \frac{8}{3} \) hours. Therefore, in one hour, B can do:
\[ \text{Work done by B in 1 hour} = \frac{3 \text{ work}}{8 \text{ hours}} = 0.375 \text{ work/hour} \]

### Step 3: Calculate the work done by C in one hour
C can do the same work as A in 8 hours, so in one hour, C can do:
\[ \text{Work done by C in 1 hour} = \frac{1 \text{ work}}{8 \text{ hours}} = 0.125 \text{ work/hour} \]

### Step 4: Calculate the total work done by A, B, and C together in one hour
Now, we can add the work done by A, B, and C in one hour:
\[ \text{Total work done in 1 hour} = \text{Work done by A} + \text{Work done by B} + \text{Work done by C} \]
\[ = 0.5 + 0.375 + 0.125 = 1 \text{ work/hour} \]

### Step 5: Calculate the time taken to complete the work together
Since A, B, and C together can complete 1 work in 1 hour, the time taken to complete the entire work is:
\[ \text{Time taken} = \frac{\text{Total work}}{\text{Total work done in 1 hour}} = \frac{1 \text{ work}}{1 \text{ work/hour}} = 1 \text{ hour} \]

Thus, the time taken for A, B, and C to complete the work together is **1 hour**.