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A complete a project in 20 days and B ca...

A 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 was completed , in how many days the project was completed?

A

18 days

B

27 days

C

26.67 days

D

16 days

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will first determine the work done by A and B individually, then calculate how long they worked together, and finally find the total time taken to complete the project. ### Step 1: Calculate the work done by A and B in one day - A can complete the project in 20 days, so A's work in one day is \( \frac{1}{20} \). - B can complete the project in 30 days, so B's work in one day is \( \frac{1}{30} \). ### Step 2: Calculate the combined work done by A and B in one day - The combined work done by A and B in one day is: \[ \text{Work done by A and B in one day} = \frac{1}{20} + \frac{1}{30} \] To add these fractions, we find a common denominator, which is 60: \[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \] Thus, \[ \text{Combined work} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \] So, A and B together can complete \( \frac{1}{12} \) of the project in one day. ### Step 3: Determine how long A and B worked together Let the total time taken to complete the project be \( x \) days. Since A quits 10 days before the project is completed, A works for \( x - 10 \) days. ### Step 4: Calculate the work done by A and B together In \( x - 10 \) days, A and B together will complete: \[ \text{Work done by A and B together} = (x - 10) \times \frac{1}{12} \] In the last 10 days, only B works, completing: \[ \text{Work done by B alone} = 10 \times \frac{1}{30} = \frac{10}{30} = \frac{1}{3} \] ### Step 5: Set up the equation for total work The total work done must equal 1 (the whole project): \[ (x - 10) \times \frac{1}{12} + \frac{1}{3} = 1 \] ### Step 6: Solve the equation First, convert \( \frac{1}{3} \) to have a common denominator of 12: \[ \frac{1}{3} = \frac{4}{12} \] Now, substitute this back into the equation: \[ (x - 10) \times \frac{1}{12} + \frac{4}{12} = 1 \] Multiply through by 12 to eliminate the fraction: \[ x - 10 + 4 = 12 \] Simplifying gives: \[ x - 6 = 12 \implies x = 18 \] ### Conclusion The project was completed in **18 days**.
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