There are three landscapers that are instructed to prune a tree. One of them alone can complete the task in 30 minutes, another is twice as fast as the first landscaper, and the thrid is a quarter as fast as the first two landscapers working together. Determine the time (in minutes) for all three landscapers working together to complete the task.

First - 30 minutes

Second - 60 minutes
Third - 112.5 minutes

Average - 67.5 minutes

^^ no idea if I'm right or not

Your answer makes no sense.

If the second worker works twice as fast than the first, then his time should be HALF of that of the first, not double the time.
so first takes 30 minutes, rate is Tree/30
the second takes 15 minutes, rate is Tree/15

combined rate is Tree/30 + Tree/15
=3Tree/30
= Tree/10

so the combined time of the first two would be 10 minutes.
since the third is a quarter as fast, then his time alone would be 40 minutes.

so the combined time would be
Tree/[Tree/30 + Tree/15 + Tree/40]
= Tree/[15Tree/120]
= 8

so it would take 8 minutes

It takes Alan and Carl 40 hours to paint a house, Bill and Carl 80 hours to paint the house, and Alan and Bill 60 hours to paint the house. How long, to the nearest minute, will it take each working alone to paint the house and how long will it take all three of them working together to paint the house?

1--The combined time of two efforts is derived from one half the harmonic mean of the two individual times or Tc = AB/(A + B), A and B being the individual times of each participant.
2--Therefore, we can write
AC/(A + C) = 40 or AC = 40A + 40C (a)
BC/(B + C) = 80 or BC = 80B + 80C (b)
AB/(A + B) = 60 or AB = 60A + 60B (c)
3--From (a) and (c), 40C/(C - 40) = 60B/(B - 60)
4--Cross multiplying, 40BC - 2400C = 60BC - 2400B or BC = 120(B - C)
5--Equating to (b) yields 120(B - C) = 80(B + C)
6--Expanding and simplifying, 40B = 200C or B = 5C
7--Substituting into (b), 5C^2 = 400C + 80C = 480C making 5C = 480 or C = 96.
8--Therefore, B = 480 and A = 68.571
9--The combined working time of three individual efforts is derived from Tc = ABC/(AB + AC + BC)
10--Therefore, the combined time for all three to paint the house is
Tc = 68.571(480)96/[(68.571x480) + (68.571x96) + 480x96)) = 36.923 hours = 36 hr - 55.377 min

Apply Tc = ABC/(AB + +AC + BC) to your problem.

To determine the time for all three landscapers working together to complete the task, we need to find their individual rates of work.

Let's assign a variable to represent the first landscaper's rate of work. Since it takes the first landscaper 30 minutes to complete the task alone, their rate of work is 1/30 of the task per minute.

Since the second landscaper is twice as fast as the first landscaper, their rate of work is 2 * (1/30) = 2/30 or 1/15 of the task per minute.

The third landscaper is a quarter as fast as the first two landscapers working together. When the first two landscapers work together, their combined rate of work is (1/30) + (1/15) = 1/20 of the task per minute. So, the third landscaper's rate of work is 1/4 * (1/20) = 1/80 of the task per minute.

To find the combined rate of work of all three landscapers, we add up their individual rates. So, the combined rate of work is:
1/30 + 1/15 + 1/80

To add the fractions, we need to find a common denominator. The least common multiple (LCM) of 30, 15, and 80 is 240.

Converting the fractions to have a common denominator of 240:
8/240 + 16/240 + 3/240

Now that the fractions have the same denominator, we can add them together:
(8 + 16 + 3)/240
= 27/240

Simplifying the fraction:
27/240 = 9/80

So, the combined rate of work for all three landscapers is 9/80 of the task per minute.

To determine the time it takes for all three landscapers working together to complete the task, we can use the formula:
Time = 1 / Rate

Therefore, the time for all three landscapers working together to complete the task is:
Time = 1 / (9/80)

Simplifying the expression:
Time = 80/9

Converting the fraction to a decimal and finding the quotient:
Time ≈ 8.89 minutes

Therefore, all three landscapers working together will take approximately 8.89 minutes to complete the task.