A and B can do a piece of work in 40 days, B and C in 30 days, and C and A in 24 days.
In what time can each finish it working alone ?

Answer

To solve the problem step by step, we will denote the amount of work done by A, B, and C in one day as \(X\), \(Y\), and \(Z\) respectively.

### Step 1: Set up the equations based on the given information

1. A and B can complete the work in 40 days:
\[ X + Y = \frac{1}{40} \quad \text{(Equation 1)} \]

2. B and C can complete the work in 30 days:
\[ Y + Z = \frac{1}{30} \quad \text{(Equation 2)} \]

3. C and A can complete the work in 24 days:
\[ Z + X = \frac{1}{24} \quad \text{(Equation 3)} \]

### Step 2: Rewrite the equations

We can rewrite the equations to express them in terms of work done per day:
- From Equation 1:
\[ X + Y = \frac{1}{40} \]

- From Equation 2:
\[ Y + Z = \frac{1}{30} \]

- From Equation 3:
\[ Z + X = \frac{1}{24} \]

### Step 3: Solve the equations

Now we will solve these equations step by step.

**Subtract Equation 2 from Equation 1:**
\[ (X + Y) - (Y + Z) = \frac{1}{40} - \frac{1}{30} \]
This simplifies to:
\[ X - Z = \frac{1}{40} - \frac{1}{30} \]
Finding a common denominator (120):
\[ X - Z = \frac{3 - 4}{120} = -\frac{1}{120} \quad \text{(Equation 4)} \]

**Add Equation 3 and Equation 4:**
\[ (Z + X) + (X - Z) = \frac{1}{24} - \frac{1}{120} \]
This simplifies to:
\[ 2X = \frac{5 - 1}{120} = \frac{4}{120} = \frac{1}{30} \]
Thus,
\[ X = \frac{1}{60} \quad \text{(A's work rate)} \]

### Step 4: Substitute \(X\) back to find \(Y\) and \(Z\)

**Substituting \(X\) into Equation 1:**
\[ \frac{1}{60} + Y = \frac{1}{40} \]
Finding a common denominator (120):
\[ \frac{2}{120} + Y = \frac{3}{120} \]
Thus,
\[ Y = \frac{3}{120} - \frac{2}{120} = \frac{1}{120} \quad \text{(B's work rate)} \]

**Substituting \(Y\) into Equation 2:**
\[ \frac{1}{120} + Z = \frac{1}{30} \]
Finding a common denominator (120):
\[ \frac{1}{120} + Z = \frac{4}{120} \]
Thus,
\[ Z = \frac{4}{120} - \frac{1}{120} = \frac{3}{120} = \frac{1}{40} \quad \text{(C's work rate)} \]

### Step 5: Calculate the time taken by each to finish the work alone

- Time taken by A to finish the work alone:
\[ \text{Time}_A = \frac{1}{X} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

- Time taken by B to finish the work alone:
\[ \text{Time}_B = \frac{1}{Y} = \frac{1}{\frac{1}{120}} = 120 \text{ days} \]

- Time taken by C to finish the work alone:
\[ \text{Time}_C = \frac{1}{Z} = \frac{1}{\frac{1}{40}} = 40 \text{ days} \]

### Final Answer:
- A can finish the work alone in 60 days.
- B can finish the work alone in 120 days.
- C can finish the work alone in 40 days.