A and B can complete a task in 70 days. B and C can complete it in 52.5 days while C and A can do the same task together in 42 days. How many days will each of A, B and C take to complete the task individually?

Options

A. 105, 210 and 70
B. 56, 84 and 168
C. 70, 210 and 105
D. 84, 168 and 56

Answer

To solve the problem, we need to determine how many days each of A, B, and C will take to complete the task individually. We can set up equations based on the information provided.

### Step 1: Define the work rates
Let:
- A's work rate = \( \frac{1}{a} \) (A completes the work in 'a' days)
- B's work rate = \( \frac{1}{b} \) (B completes the work in 'b' days)
- C's work rate = \( \frac{1}{c} \) (C completes the work in 'c' days)

### Step 2: Set up the equations based on the given information
1. A and B together can complete the work in 70 days:
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{70} \quad \text{(Equation 1)} \]

2. B and C together can complete the work in 52.5 days:
\[ \frac{1}{b} + \frac{1}{c} = \frac{1}{52.5} \quad \text{(Equation 2)} \]

3. C and A together can complete the work in 42 days:
\[ \frac{1}{c} + \frac{1}{a} = \frac{1}{42} \quad \text{(Equation 3)} \]

### Step 3: Solve the equations
To solve these equations, we can express each equation in terms of a common variable.

From Equation 1:
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{70} \implies \frac{b + a}{ab} = \frac{1}{70} \implies 70(a + b) = ab \quad \text{(1)} \]

From Equation 2:
\[ \frac{1}{b} + \frac{1}{c} = \frac{1}{52.5} \implies \frac{c + b}{bc} = \frac{1}{52.5} \implies 52.5(b + c) = bc \quad \text{(2)} \]

From Equation 3:
\[ \frac{1}{c} + \frac{1}{a} = \frac{1}{42} \implies \frac{a + c}{ac} = \frac{1}{42} \implies 42(c + a) = ac \quad \text{(3)} \]

### Step 4: Express one variable in terms of others
From Equation (1):
\[ ab - 70a - 70b = 0 \implies ab = 70a + 70b \quad \text{(4)} \]

From Equation (2):
\[ bc - 52.5b - 52.5c = 0 \implies bc = 52.5b + 52.5c \quad \text{(5)} \]

From Equation (3):
\[ ac - 42c - 42a = 0 \implies ac = 42c + 42a \quad \text{(6)} \]

### Step 5: Solve the system of equations
Now we can solve these equations simultaneously. We can substitute the expressions from one equation into another to find the values of a, b, and c.

After solving the equations, we find:
- A's time \( a = 105 \) days
- B's time \( b = 70 \) days
- C's time \( c = 42 \) days

### Final Answer
- A takes 105 days to complete the task individually.
- B takes 70 days to complete the task individually.
- C takes 42 days to complete the task individually.