**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?

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Accepted Answer

**Solution:**

1. Let the total time to complete the project be $x$ days.

2. Work done by **A** in $(x - 10)$ days = $\frac{x - 10}{20}$ (since A completes the project in 20 days)

3. Work done by **B** in $x$ days = $\frac{x}{30}$ (since B completes the project in 30 days)

4. Total work done by A and B together equals 1 (the whole project):
   $$\frac{x - 10}{20} + \frac{x}{30} = 1$$

5. Multiply both sides by 60 (LCM of 20 and 30):
   $$3(x - 10) + 2x = 60$$

6. Simplify:
   $$3x - 30 + 2x = 60$$
   $$5x - 30 = 60$$

7. Solve for $x$:
   $$5x = 90$$
   $$x = 18$$

**Answer:** The project will be completed in $18$ days.

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