13. A and B together can clean a lab in 4 days. Independently, A can clean the lab in 20 days. How many days
will it take for B to independently complete the task?
1. 4
2. 5
3. 16
4. 12.1

Calculating the Time for B to Clean the Lab Independently

In this problem, we are given that A and B together can clean a lab in 4 days, and A can independently clean the lab in 20 days. We are asked to determine how many days it will take for B to independently complete the task.

Step-by-Step Solution:

1. Work Rate of A and B Together:

   • The combined rate of A and B cleaning the lab together is 14\frac{1}{4}41​ of the lab per day, since they can clean it in 4 days.

2. Work Rate of A:

   • A can clean the lab in 20 days, so A’s work rate is 120\frac{1}{20}201​ of the lab per day.

3. Work Rate of B:

   • Let B’s work rate be 1x\frac{1}{x}x1​, where $x$ is the number of days it takes for B to clean the lab independently.

4. Equation for Combined Work Rate:

   • The combined work rate of A and B is the sum of their individual work rates:

     14=120+1x\frac{1}{4} = \frac{1}{20} + \frac{1}{x}41​=201​+x1​

5. Solve for $x$:

   • Rearranging the equation:

     1x=14−120\frac{1}{x} = \frac{1}{4} – \frac{1}{20}x1​=41​−201​

     To subtract the fractions, find a common denominator (20):

     1x=520−120=420=15\frac{1}{x} = \frac{5}{20} – \frac{1}{20} = \frac{4}{20} = \frac{1}{5}x1​=205​−201​=204​=51​

   • Therefore:

     $$x=5$$

✅ Correct Answer:

(2) 5

Conclusion:

It will take B 5 days to clean the lab independently. This type of problem is a classic example of work rate problems, which can be solved using the concept of combined work rates. Understanding how to break down and combine rates is essential in many mathematical applications such as project management and teamwork scenarios.