A can complete a project in 20 days and B can complete the same project in 30 days. If A and B start working on the project together and A quits 10 days before the project is completed, in how many days will the project be completed?

Options

A. 18
B. 27
C. 26.67
D. 16

Answer

To solve the problem step by step, we need to determine how many days it will take for A and B to complete the project together, given that A quits 10 days before the project is finished.

### Step 1: Determine the work rates of A and B
- A can complete the project in 20 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{20} \text{ of the project per day} \]

- B can complete the project in 30 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{30} \text{ of the project per day} \]

### Step 2: Combine their work rates
When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{30} \]
To add these fractions, we need a common denominator, which is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Thus,
\[ \text{Combined work rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]
This means together, A and B can complete \(\frac{1}{12}\) of the project in one day.

### Step 3: Set up the equation for total work
Let \(x\) be the total number of days the project takes to complete. Since A quits 10 days before the project is completed, A works for \(x - 10\) days, and B works for all \(x\) days.

The work done by A in \(x - 10\) days is:
\[ \text{Work done by A} = \frac{1}{20} \times (x - 10) \]
The work done by B in \(x\) days is:
\[ \text{Work done by B} = \frac{1}{30} \times x \]

### Step 4: Write the equation for total work
The total work done by A and B together must equal 1 (the whole project):
\[ \frac{1}{20}(x - 10) + \frac{1}{30}x = 1 \]

### Step 5: Solve the equation
To solve the equation, we first find a common denominator for the fractions, which is 60:
\[ \frac{3}{60}(x - 10) + \frac{2}{60}x = 1 \]
Multiplying through by 60 to eliminate the fractions:
\[ 3(x - 10) + 2x = 60 \]
Expanding the equation:
\[ 3x - 30 + 2x = 60 \]
Combining like terms:
\[ 5x - 30 = 60 \]
Adding 30 to both sides:
\[ 5x = 90 \]
Dividing by 5:
\[ x = 18 \]

### Final Answer
The project will be completed in **18 days**.

---