(A + B) can do a piece of work in 12 days. (B + C) can do same work in `6(2)/(3)` days. A work for 3 days, B works for 4 days and C completed the remaining work in 7 days. In how many days will A, B and C do this work alone ?

Options

A. 18,15.,20
B. 30,20,10
C. 10,15,25
D. 15,10,25

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, and C based on the given information.

### Step 1: Determine the total work done by A + B and B + C
- A + B can complete the work in 12 days.
- B + C can complete the work in \(6 \frac{2}{3}\) days, which is equal to \(\frac{20}{3}\) days.

### Step 2: Calculate the efficiencies of A + B and B + C
- The efficiency of A + B:
\[ \text{Efficiency of A + B} = \frac{\text{Total Work}}{\text{Time}} = \frac{1}{12} \]

- The efficiency of B + C:
\[ \text{Efficiency of B + C} = \frac{\text{Total Work}}{\text{Time}} = \frac{3}{20} \]

### Step 3: Find a common total work
To find a common total work, we can use the least common multiple (LCM) of the denominators:
- LCM of 12 and \(\frac{20}{3}\) is 60.

### Step 4: Calculate the work done by A + B and B + C
- If the total work is 60:
- Work done by A + B in 12 days:
\[ \text{Efficiency of A + B} = \frac{60}{12} = 5 \text{ units/day} \]

- Work done by B + C in \(6 \frac{2}{3}\) days:
\[ \text{Efficiency of B + C} = \frac{60}{\frac{20}{3}} = 9 \text{ units/day} \]

### Step 5: Set up equations for A, B, and C
Let the efficiencies of A, B, and C be \(a\), \(b\), and \(c\) respectively. We have:
1. \(a + b = 5\)
2. \(b + c = 9\)

### Step 6: Calculate the work done by A, B, and C
- A works for 3 days:
\[ \text{Work done by A} = 3a \]

- B works for 4 days:
\[ \text{Work done by B} = 4b \]

- C completes the remaining work in 7 days:
\[ \text{Work done by C} = 7c \]

### Step 7: Total work equation
The total work done by A, B, and C is equal to the total work:
\[ 3a + 4b + 7c = 60 \]

### Step 8: Substitute values from previous equations
From \(a + b = 5\), we can express \(a\) as:
\[ a = 5 - b \]

Substituting \(a\) into the total work equation:
\[ 3(5 - b) + 4b + 7c = 60 \]
\[ 15 - 3b + 4b + 7c = 60 \]
\[ 15 + b + 7c = 60 \]
\[ b + 7c = 45 \quad \text{(Equation 1)} \]

From \(b + c = 9\), we can express \(c\) as:
\[ c = 9 - b \]

Substituting \(c\) into Equation 1:
\[ b + 7(9 - b) = 45 \]
\[ b + 63 - 7b = 45 \]
\[ -6b + 63 = 45 \]
\[ -6b = 45 - 63 \]
\[ -6b = -18 \]
\[ b = 3 \]

### Step 9: Find values of a and c
Using \(b = 3\):
- From \(a + b = 5\):
\[ a + 3 = 5 \implies a = 2 \]

- From \(b + c = 9\):
\[ 3 + c = 9 \implies c = 6 \]

### Step 10: Calculate the number of days each can complete the work alone
- Days taken by A:
\[ \text{Days by A} = \frac{60}{a} = \frac{60}{2} = 30 \text{ days} \]

- Days taken by B:
\[ \text{Days by B} = \frac{60}{b} = \frac{60}{3} = 20 \text{ days} \]

- Days taken by C:
\[ \text{Days by C} = \frac{60}{c} = \frac{60}{6} = 10 \text{ days} \]

### Final Answer
A can complete the work in 30 days, B in 20 days, and C in 10 days.