A and B can do a piece of work in 18 days. B and C in 24 days, C and A in 36 days. Find the time in whichA, Band C working together can finish the work.

Options

A. 8
B. 16
C. 24
D. 36

Answer

To solve the problem step by step, we need to find the time taken by A, B, and C working together based on the information provided about their pairwise work rates.

### Step 1: Understand the given information
- A and B can complete the work in 18 days.
- B and C can complete the work in 24 days.
- C and A can complete the work in 36 days.

### Step 2: Calculate the work rates of each pair
We can express the work done by each pair in terms of their rates:
- Work done by A and B in one day = \( \frac{1}{18} \)
- Work done by B and C in one day = \( \frac{1}{24} \)
- Work done by C and A in one day = \( \frac{1}{36} \)

### Step 3: Set up equations for the work rates
Let:
- Rate of A = \( a \)
- Rate of B = \( b \)
- Rate of C = \( c \)

From the information given, we can set up the following equations:
1. \( a + b = \frac{1}{18} \) (1)
2. \( b + c = \frac{1}{24} \) (2)
3. \( c + a = \frac{1}{36} \) (3)

### Step 4: Solve the equations
We can add all three equations together:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{18} + \frac{1}{24} + \frac{1}{36} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{18} + \frac{1}{24} + \frac{1}{36} \]

### Step 5: Find the common denominator
The least common multiple (LCM) of 18, 24, and 36 is 72. We convert each fraction:
- \( \frac{1}{18} = \frac{4}{72} \)
- \( \frac{1}{24} = \frac{3}{72} \)
- \( \frac{1}{36} = \frac{2}{72} \)

Now, we can add them:
\[ \frac{4}{72} + \frac{3}{72} + \frac{2}{72} = \frac{9}{72} = \frac{1}{8} \]

### Step 6: Substitute back to find \( a + b + c \)
Thus, we have:
\[ 2(a + b + c) = \frac{1}{8} \]
Dividing both sides by 2 gives:
\[ a + b + c = \frac{1}{16} \]

### Step 7: Find the time taken by A, B, and C together
The time taken by A, B, and C working together is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{a + b + c} = \frac{1}{\frac{1}{16}} = 16 \text{ days} \]

### Final Answer
A, B, and C together can finish the work in **16 days**.

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