A and B working together can complete a piece of work in 6 days, B and C in 10 days, C and A in `7(1)/(2)` days. The number of days required by A,B and C respectively to complete the work individually is

Options

A. 15,20,30
B. 10,20,30
C. 10,15,30
D. 20,15,10

Explanation

To solve the problem step by step, we will determine the individual efficiencies of A, B, and C based on the information given about their combined work rates.

### Step 1: Understand the combined work rates
- A and B can complete the work in 6 days.
- B and C can complete the work in 10 days.
- C and A can complete the work in 7.5 days (which is 7 and 1/2 days).

### Step 2: Calculate the work done per day by each pair
1. **A and B together:**
\[ \text{Work done in 1 day} = \frac{1}{6} \text{ of the work} = 5 \text{ units (assuming total work is 30 units)} \]

2. **B and C together:**
\[ \text{Work done in 1 day} = \frac{1}{10} \text{ of the work} = 3 \text{ units} \]

3. **C and A together:**
\[ \text{Work done in 1 day} = \frac{1}{7.5} \text{ of the work} = \frac{30}{15} = 4 \text{ units} \]

### Step 3: Set up equations based on the work done
Let:
- A's efficiency = \( a \)
- B's efficiency = \( b \)
- C's efficiency = \( c \)

From the work done per day:
1. \( a + b = 5 \) (Equation 1)
2. \( b + c = 3 \) (Equation 2)
3. \( c + a = 4 \) (Equation 3)

### Step 4: Solve the equations
**Add all three equations:**
\[ (a + b) + (b + c) + (c + a) = 5 + 3 + 4 \]
This simplifies to:
\[ 2a + 2b + 2c = 12 \]
Dividing through by 2 gives:
\[ a + b + c = 6 \quad \text{(Equation 4)} \]

### Step 5: Substitute to find individual efficiencies
1. From Equation 1: \( b = 5 - a \)
2. Substitute \( b \) in Equation 2:
\[ (5 - a) + c = 3 \implies c = 3 - (5 - a) = a - 2 \]
3. Substitute \( c \) in Equation 3:
\[ (a - 2) + a = 4 \implies 2a - 2 = 4 \implies 2a = 6 \implies a = 3 \]
4. Now substitute \( a \) back to find \( b \) and \( c \):
- From Equation 1: \( b = 5 - 3 = 2 \)
- From \( c = a - 2 \): \( c = 3 - 2 = 1 \)

### Step 6: Determine the number of days each person takes to complete the work individually
- Total work = 30 units
1. Days taken by A:
\[ \text{Days} = \frac{30}{a} = \frac{30}{3} = 10 \text{ days} \]
2. Days taken by B:
\[ \text{Days} = \frac{30}{b} = \frac{30}{2} = 15 \text{ days} \]
3. Days taken by C:
\[ \text{Days} = \frac{30}{c} = \frac{30}{1} = 30 \text{ days} \]

### Final Answer:
- A takes 10 days,
- B takes 15 days,
- C takes 30 days.