A can do a piece of work in 10 days, B in 12 days and C in 15 days. All begin together but A leaves the work after 2 days and B leaves 3 days befor the work is finished. How long did the work last ?

Answer

To solve the problem step by step, we will first determine the efficiency of each worker, then calculate the total work done, and finally find out how long the work lasted.

### Step 1: Determine the efficiency of A, B, and C
- A can complete the work in 10 days, so A's efficiency = 1/10 of the work per day.
- B can complete the work in 12 days, so B's efficiency = 1/12 of the work per day.
- C can complete the work in 15 days, so C's efficiency = 1/15 of the work per day.

### Step 2: Find the least common multiple (LCM) of the days
To find a common measure for their efficiencies, we calculate the LCM of 10, 12, and 15.
- LCM(10, 12, 15) = 60

### Step 3: Convert efficiencies to a common scale
Now we convert their efficiencies based on the LCM:
- A's efficiency = 60/10 = 6 units of work per day
- B's efficiency = 60/12 = 5 units of work per day
- C's efficiency = 60/15 = 4 units of work per day

### Step 4: Calculate the total work
The total work can be calculated as:
Total work = Efficiency × Time = 60 units (as calculated from A's work).

### Step 5: Calculate work done in the first 2 days
In the first 2 days, all three work together:
- Total efficiency of A, B, and C = 6 + 5 + 4 = 15 units/day
- Work done in 2 days = 2 × 15 = 30 units

### Step 6: Remaining work after 2 days
Remaining work = Total work - Work done in 2 days = 60 - 30 = 30 units

### Step 7: Determine the time taken by B and C
B leaves 3 days before the work is finished, which means we need to find out how long B and C worked together before B left.

Let T be the total time taken to finish the work. Since B leaves 3 days before the work is finished, B and C work together for (T - 3) days.

### Step 8: Work done by B and C together
During the time B and C work together:
- Work done by B and C in (T - 3) days = (5 + 4) × (T - 3) = 9(T - 3) units

### Step 9: Work done by C alone
After B leaves, C works alone for 3 days:
- Work done by C in 3 days = 4 × 3 = 12 units

### Step 10: Set up the equation
Total work done = Work done by A + Work done by B and C + Work done by C alone:
\[ 30 + 9(T - 3) + 12 = 60 \]

### Step 11: Solve the equation
Expanding the equation:
\[ 30 + 9T - 27 + 12 = 60 \]
\[ 9T + 15 = 60 \]
\[ 9T = 60 - 15 \]
\[ 9T = 45 \]
\[ T = 5 \]

### Step 12: Calculate total time of work
Total time of work = Time A worked + Time B and C worked together + Time C worked alone
\[ = 2 + (T - 3) + 3 \]
\[ = 2 + (5 - 3) + 3 \]
\[ = 2 + 2 + 3 = 7 \text{ days} \]

### Final Answer
The total duration of the work lasted **7 days**.

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