__Problem 1:__

It takes 1.5 hours for Tim to mow the lawn. Linda can mow the same lawn in 2 hours. How long will it take John and Linda, work together, to mow the lawn?

__Solution to Problem 1:__

- We first calculate the rate of work of John and Linda

John: 1 / 1.5 and Linda 1 / 2

- Let t be the time for John and Linda to mow the Lawn. The work done by John alone is given by

t * (1 / 1.5)

- The work done by Linda alone is given by

t * (1 / 2)

- When the two work together, their work will be added. Hence

t * (1 / 1.5) + t * (1 / 2) = 1

- Multiply all terms by 6

6 (t * (1 / 1.5) + t * (1 / 2) ) = 6

- and simplify

4 t + 3 t = 6

- Solve for t

t = 6 / 7 hours = 51.5 minutes.

__Problem 2:__
It takes 6 hours for pump A, used alone, to fill a tank of water. Pump B used alone takes 8 hours to fill the same tank. We want to use three pumps: A, B and another pump C to fill the tank in 2 hours. What should be the rate of pump C? How long would it take pump C, used alone, to fill the tank?

__Solution to Problem 2:__

- The rates of pumps A and B can be calculated as follows:

A: 1 / 6 and B: 1 / 8

- Let R be the rate of pump C. When working together for 2 hours, we have

2 ( 1 / 6 + 1 / 8 + R ) = 1

- Solve for R

R = 1 / 4.8 , rate of pump C.

- Let t be the time it takes pump C, used alone, to fill the tank. Hence

t * (1 / 4.8) = 1

- Solve for t

t = 4.8 hours , the time it takes pump C to fill the tank.

__Problem 3:__
A tank can be filled by pipe A in 5 hours and by pipe B in 8 hours, each pump working on its own. When the tank is full and a drainage hole is open, the water is drained in 20 hours. If initially the tank was empty and someone started the two pumps together but left the drainage hole open, how long does it take for the tank to be filled?

__Solution to Problem 3:__

- Let's first find the rates of the pumps and the drainage hole

pump A: 1 / 5 , pump B: 1 / 8 , drainage hole: 1 / 20

- Let t be the time for the pumps to fill the tank. The pumps ,
**add** water into the tank however the drainage hole **drains** water out of the tank, hence

t ( 1 / 5 + 1 / 8 - 1 / 20) = 1

- Solve for t

t = 3.6 hours.

__Problem 4:__
A swimming pool can be filled by pipe A in 3 hours and by pipe B in 6 hours, each pump working on its own. At 9 am pump A is started. At what time will the swimming pool be filled if pump B is started at 10 am?

__Solution to Problem 4:__

- the rates of the two pumps are

pump A: 1 / 3 , pump B: 1 / 6

- Working together, If pump A works for t hours then pump B works t - 1 hours since it started 1 hour late. Hence

t * (1 / 3) + (t - 1) * (1 / 6) = 1

- Solve for t

t = 7 / 3 hours = 2.3 hours = 2 hours 20 minutes.

- The swimming pool will be filled at

9 + 2:20 = 11:20

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