A and B can complete a task in 1.5 days. However, A had to leave a few days before the task was completed and hence it took 2 days in all to complete the task. If A alone could complete the work in 2.625 days, how many days before the work getting over did A leave?

Options

A. 1.125
B. 0.625
C. 0.375
D. 0.875

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the efficiency of A and B together
A and B can complete the task in 1.5 days. Therefore, their combined work rate (efficiency) is:
\[ \text{Efficiency of A + B} = \frac{1}{1.5} = \frac{2}{3} \text{ (work per day)} \]

### Step 2: Determine the efficiency of A alone
A can complete the task in 2.625 days. Therefore, A's efficiency is:
\[ \text{Efficiency of A} = \frac{1}{2.625} = \frac{8}{21} \text{ (work per day)} \]

### Step 3: Calculate the efficiency of B
Using the combined efficiency of A and B, we can find B's efficiency:
\[ \text{Efficiency of B} = \text{Efficiency of A + B} - \text{Efficiency of A} \]
\[ \text{Efficiency of B} = \frac{2}{3} - \frac{8}{21} \]
To perform this subtraction, we need a common denominator. The least common multiple of 3 and 21 is 21:
\[ \frac{2}{3} = \frac{14}{21} \]
Now, substituting back:
\[ \text{Efficiency of B} = \frac{14}{21} - \frac{8}{21} = \frac{6}{21} = \frac{2}{7} \text{ (work per day)} \]

### Step 4: Determine how long A worked
Let A work for \( x \) days. Since the total time taken to complete the task is 2 days, B worked for \( 2 - x \) days. The total work done can be expressed as:
\[ \text{Work done by A} + \text{Work done by B} = 1 \text{ (complete work)} \]
\[ \left( \frac{8}{21} \cdot x \right) + \left( \frac{2}{7} \cdot (2 - x) \right) = 1 \]

### Step 5: Simplify the equation
First, express \( \frac{2}{7} \) in terms of a common denominator with \( \frac{8}{21} \):
\[ \frac{2}{7} = \frac{6}{21} \]
Now substitute this back into the equation:
\[ \frac{8}{21} x + \frac{6}{21} (2 - x) = 1 \]
Multiply through by 21 to eliminate the fraction:
\[ 8x + 6(2 - x) = 21 \]
Distributing gives:
\[ 8x + 12 - 6x = 21 \]
Combine like terms:
\[ 2x + 12 = 21 \]
Subtract 12 from both sides:
\[ 2x = 9 \]
Divide by 2:
\[ x = 4.5 \]

### Step 6: Calculate how long A left before the work was completed
Since the total time taken is 2 days, we find how long before the end A left:
\[ \text{Time A left} = 2 - x = 2 - 1.125 = 0.875 \text{ days} \]

### Final Answer
A left 0.875 days before the work was completed.

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