Algebra and problem solving---Work?

Question

I would really appreciate it if you people can provide the steps and explanation =)

1. Alec can dig a ditch in 4 hours. Ivan can dig the same ditch in 3 hours. How long would it take them to dig it together?

2. If Susana can type a paper in 5 hours and together she and Sophia can type it in two hours, how long would it take Sophia to type the same paper alone?

3. Two boys paint a room. Leo can paint it alone in 5 days, Alec in 8 days. They start to paint it together, but after 2 days Alec gets bored and Leo finished the job alone. How long does it take him to finish?

4. A swimming pool can be filled by an inlet pipe in 10 hours and emptied by an outlet pipe in 12 hours. One day the pool is empty and the owner opens the inlet pipe to fill the pool. But he forgets to close the outlet. With both pipes open, how long will it take to fill the pool?

5. Mary, Sue and Bill work at a hotel. If each worked alone, it would take Mary 10 hours, Sue 8 hours, and Bill 12 hours to clean the hotel. One day Mary came to work early and she cleaned for 2 hours when Sue and Bill arrived and all three finished the job. How long did they take to finish?

Answer 1

All of these are the same type of problem.  I'll take the first two, but you should be able to get the rest.

The key to them, in my experience, is to look at them as rate problems.  For instance:

1)  Alec digs 1/4 ditch per hour.  Ivan digs 1/3 ditch per hour.  Together they dig 1/4 + 1/3 = 7/12 ditch per hour.  So, it'll take 12/7 of an hour for the two of them together.  12/7 = 1 hour 42.85 minutes.

2)  Susana types 1/5 of a paper per hour.  Sophia types 1/x paper per hour.  Together, they type 1/2 paper per hour. So,

1/5 + 1/x = 1/2

1/x = 1/2 - 1/5 = 3/10

x = 10/3 hours per paper = 3 hour 20 minutes.

See the pattern?

Answer 2

1. A's rate is 1/4 ditch per hour (3/12)

I's rate is 1/3 ditch per hour (4/12)

Together their rate is 7/12 ditch per hour.

rate=ditches dug/time

time = ditch/ rate = 1/(7/12) = 12/7 hours

12/7*60 =720/7 = 102.86minutes

Time to dig one ditch together = approximately 1 hour, 43 minutes.

2. If Susana can type a paper in 5 hours and together she and Sophia can type it in two hours, how long would it take Sophia to type the same paper alone?

Susana types at 1/5 paper per hour (2/10)

Together they type at 1/2 paper per hour (5/10)

Sophia's share must be 3/10 paper per hour.

1/(3/10) = 10/3 hours per paper = 600/3 minutes = 200 minutes = 3 1/3 hours.

Sophia would take 3 1/3 hours on her own.