A and B together can complete a task in 1.2 days. However, if A works alone, completes half the job and leaves and 8 then B works alone and completes the rest of the work, it takes 2.5 days in all to complete the work. If B is more efficient than A, how many days would it have taken B to do the work by herself?

Options

A. `1.5`
B. `1.8`
C. `2.0`
D. `2.2`

Answer

To solve the problem step by step, we will use the information given in the question about A and B's work rates and the total time taken to complete the task.

### Step 1: Define Variables
Let:
- A's time to complete the work alone = \( x \) days
- B's time to complete the work alone = \( y \) days

### Step 2: Write the Work Rate Equation
From the information that A and B together can complete the task in 1.2 days, we can express their combined work rate:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{1.2} \]
This can be rewritten as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{5}{6} \]
Multiplying through by \( xy \) gives:
\[ y + x = \frac{5xy}{6} \quad \text{(Equation 1)} \]

### Step 3: Analyze the Work Done
A completes half the job and then leaves. The time taken by A to complete half the job is:
\[ \frac{x}{2} \text{ days} \]
After A leaves, B completes the remaining half of the job. The time taken by B to complete half the job is:
\[ \frac{y}{2} \text{ days} \]
The total time taken to complete the work is given as 2.5 days:
\[ \frac{x}{2} + \frac{y}{2} = 2.5 \]
Multiplying through by 2 gives:
\[ x + y = 5 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + y = \frac{5xy}{6} \)
2. \( x + y = 5 \)

From Equation 2, we can substitute \( x + y \) in Equation 1:
\[ 5 = \frac{5xy}{6} \]
Multiplying both sides by 6 gives:
\[ 30 = 5xy \]
Dividing by 5 gives:
\[ xy = 6 \quad \text{(Equation 3)} \]

### Step 5: Substitute and Solve for x and y
Now we can use Equation 2 and Equation 3 to find \( x \) and \( y \). From Equation 2:
\[ y = 5 - x \]
Substituting this into Equation 3:
\[ x(5 - x) = 6 \]
Expanding gives:
\[ 5x - x^2 = 6 \]
Rearranging gives:
\[ x^2 - 5x + 6 = 0 \]

### Step 6: Factor the Quadratic
Factoring the quadratic:
\[ (x - 2)(x - 3) = 0 \]
Thus, \( x = 2 \) or \( x = 3 \).

### Step 7: Find Corresponding y Values
Using \( y = 5 - x \):
- If \( x = 2 \), then \( y = 3 \).
- If \( x = 3 \), then \( y = 2 \).

### Step 8: Determine the More Efficient Worker
Since it is given that B is more efficient than A, we conclude:
- If \( y = 3 \) (B takes 3 days), then \( x = 2 \) (A takes 2 days).
- If \( y = 2 \) (B takes 2 days), then \( x = 3 \) (A takes 3 days).

Thus, B must take 2 days to complete the work alone.

### Final Answer
B would take **2 days** to complete the work by herself.

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