A and B can complete a task in 50 days. B and C can complete it in 37.5 days while C and A can do the same task together in 30 days. How many days will A. B and C each take to complete the task individually?

Options

A. 40, 60 and 120
B. 60,120 and 40
C. 75, 150 and 50
D. 50, 150 and 75

Answer

To solve the problem, we need to find out how many days A, B, and C will take to complete the task individually based on the information given about their combined efficiencies.

### Step 1: Define the efficiencies
Let’s denote:
- The efficiency of A as \( a \) (units of work per day)
- The efficiency of B as \( b \) (units of work per day)
- The efficiency of C as \( c \) (units of work per day)

### Step 2: Set up the equations based on the given information
From the problem, we have:
1. A and B can complete the task in 50 days:
\[ a + b = \frac{150}{50} = 3 \quad \text{(units of work per day)} \]
2. B and C can complete the task in 37.5 days:
\[ b + c = \frac{150}{37.5} = 4 \quad \text{(units of work per day)} \]
3. C and A can complete the task in 30 days:
\[ c + a = \frac{150}{30} = 5 \quad \text{(units of work per day)} \]

### Step 3: Write the system of equations
We now have the following system of equations:
1. \( a + b = 3 \) (1)
2. \( b + c = 4 \) (2)
3. \( c + a = 5 \) (3)

### Step 4: Solve the equations
To find the individual efficiencies, we can add all three equations:
\[ (a + b) + (b + c) + (c + a) = 3 + 4 + 5 \]
This simplifies to:
\[ 2a + 2b + 2c = 12 \]
Dividing the entire equation by 2 gives:
\[ a + b + c = 6 \quad \text{(4)} \]

### Step 5: Substitute to find individual efficiencies
Now, we can substitute equation (4) into the individual equations to find \( a \), \( b \), and \( c \).

From equation (1):
\[ c = 6 - (a + b) = 6 - 3 = 3 \]

From equation (2):
\[ a = 6 - (b + c) = 6 - 4 = 2 \]

From equation (3):
\[ b = 6 - (c + a) = 6 - 5 = 1 \]

### Step 6: Calculate the number of days for each person
Now that we have the efficiencies:
- \( a = 2 \) units/day
- \( b = 1 \) unit/day
- \( c = 3 \) units/day

We can find out how many days each will take to complete the task alone:
- A will take:
\[ \text{Days for A} = \frac{150}{a} = \frac{150}{2} = 75 \text{ days} \]
- B will take:
\[ \text{Days for B} = \frac{150}{b} = \frac{150}{1} = 150 \text{ days} \]
- C will take:
\[ \text{Days for C} = \frac{150}{c} = \frac{150}{3} = 50 \text{ days} \]

### Final Answer:
- A will take 75 days,
- B will take 150 days,
- C will take 50 days.