A and B working together, can do a piece of work in `4(1)/(2)` hours. B and C working together can do it in 3 hours. C and A working together can do it in `2(1)/(4)` hours. All of them begin the work at the same time. Find how much time they will take to finish the piece of work.

Options

A. `3` hours
B. `2` hours
C. `2.5` hours
D. `3.25` hours

Answer

To solve the problem, we need to find the individual work rates of A, B, and C, and then determine how long they will take to finish the work together.

### Step 1: Convert the mixed fractions to improper fractions.
- A and B can do the work in \(4 \frac{1}{2}\) hours, which is \( \frac{9}{2} \) hours.
- B and C can do the work in \(3\) hours, which is \(3\) hours.
- C and A can do the work in \(2 \frac{1}{4}\) hours, which is \( \frac{9}{4} \) hours.

### Step 2: Calculate the work rates of each pair.
The work done by each pair in one hour can be calculated as follows:
- Work done by A and B in one hour = \( \frac{1}{\frac{9}{2}} = \frac{2}{9} \)
- Work done by B and C in one hour = \( \frac{1}{3} \)
- Work done by C and A in one hour = \( \frac{1}{\frac{9}{4}} = \frac{4}{9} \)

### Step 3: Set up equations for the individual work rates.
Let the work rates of A, B, and C be \(a\), \(b\), and \(c\) respectively.
From the work rates calculated:
1. \( a + b = \frac{2}{9} \) (Equation 1)
2. \( b + c = \frac{1}{3} \) (Equation 2)
3. \( c + a = \frac{4}{9} \) (Equation 3)

### Step 4: Solve the equations.
From Equation 2, we can express \(c\) in terms of \(b\):
\[ c = \frac{1}{3} - b \]

Substituting \(c\) into Equation 3:
\[ \left(\frac{1}{3} - b\right) + a = \frac{4}{9} \]
\[ a - b = \frac{4}{9} - \frac{1}{3} \]
\[ a - b = \frac{4}{9} - \frac{3}{9} = \frac{1}{9} \]
So, we have:
\[ a = b + \frac{1}{9} \] (Equation 4)

Now substituting \(a\) from Equation 4 into Equation 1:
\[ (b + \frac{1}{9}) + b = \frac{2}{9} \]
\[ 2b + \frac{1}{9} = \frac{2}{9} \]
\[ 2b = \frac{2}{9} - \frac{1}{9} = \frac{1}{9} \]
\[ b = \frac{1}{18} \]

Now substitute \(b\) back into Equation 4 to find \(a\):
\[ a = \frac{1}{18} + \frac{1}{9} = \frac{1}{18} + \frac{2}{18} = \frac{3}{18} = \frac{1}{6} \]

Now substitute \(b\) into Equation 2 to find \(c\):
\[ c = \frac{1}{3} - \frac{1}{18} = \frac{6}{18} - \frac{1}{18} = \frac{5}{18} \]

### Step 5: Calculate the total work rate when A, B, and C work together.
The total work rate when A, B, and C work together is:
\[ a + b + c = \frac{1}{6} + \frac{1}{18} + \frac{5}{18} \]
To add these fractions, we need a common denominator, which is 18:
\[ = \frac{3}{18} + \frac{1}{18} + \frac{5}{18} = \frac{9}{18} = \frac{1}{2} \]

### Step 6: Calculate the time taken to finish the work together.
If A, B, and C can do \(\frac{1}{2}\) of the work in one hour, they will take:
\[ \text{Time} = \frac{1}{\frac{1}{2}} = 2 \text{ hours} \]

### Final Answer:
A, B, and C together will take **2 hours** to finish the piece of work.

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