A, B and C can complete a piece of-work in 15, 30 and 40 days respectively. They started the work together and A left 2 days before the completion of the work and B left 4 days before the completion of the work. In how many days was the work completed?

Options

A. `7 3/10`
B. `10 2/15`
C. `10 7/30`
D. none of these

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C, and then calculate how long it took to complete the work given the conditions.

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 15 days, so A's work rate is \( \frac{1}{15} \) of the work per day.
- B can complete the work in 30 days, so B's work rate is \( \frac{1}{30} \) of the work per day.
- C can complete the work in 40 days, so C's work rate is \( \frac{1}{40} \) of the work per day.

### Step 2: Find the combined work rate of A, B, and C
To find the combined work rate, we need to find a common denominator for their work rates:
- The LCM of 15, 30, and 40 is 120.
- A's work rate in terms of units of work is \( \frac{120}{15} = 8 \) units per day.
- B's work rate in terms of units of work is \( \frac{120}{30} = 4 \) units per day.
- C's work rate in terms of units of work is \( \frac{120}{40} = 3 \) units per day.

Thus, the combined work rate of A, B, and C is:
\[ 8 + 4 + 3 = 15 \text{ units per day} \]

### Step 3: Determine the work done in the last days
Let \( x \) be the total number of days taken to complete the work. According to the problem:
- A left 2 days before completion, so A worked for \( x - 2 \) days.
- B left 4 days before completion, so B worked for \( x - 4 \) days.
- C worked for all \( x \) days.

### Step 4: Calculate the total work done
The total work done can be expressed as:
\[ \text{Work done by A} = 8(x - 2) \]
\[ \text{Work done by B} = 4(x - 4) \]
\[ \text{Work done by C} = 3x \]

The total work done is equal to the total work (120 units):
\[ 8(x - 2) + 4(x - 4) + 3x = 120 \]

### Step 5: Simplify the equation
Expanding the equation:
\[ 8x - 16 + 4x - 16 + 3x = 120 \]
Combining like terms:
\[ (8x + 4x + 3x) - 32 = 120 \]
\[ 15x - 32 = 120 \]

### Step 6: Solve for \( x \)
Adding 32 to both sides:
\[ 15x = 152 \]
Dividing by 15:
\[ x = \frac{152}{15} \approx 10.13 \]

### Step 7: Determine the total days taken
Since \( x \) must be a whole number, we round up to the nearest whole number, which gives us \( x = 11 \) days.

### Final Answer
The total number of days taken to complete the work is **11 days**.

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