SOLUTION: The head gardener can mow the lawns in the city park twice as fast as his assistant. Working together, they can complete the job in 1 1/3 hours. How long would it take the head ga

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Question 1196109: The head gardener can mow the lawns in the city park twice as fast as his assistant. Working together, they can complete the job in 1 1/3 hours.
How long would it take the head gardener working alone?
Found 4 solutions by ikleyn, josgarithmetic, greenestamps, math_tutor2020:
Answer by ikleyn(52818)   (Show Source): You can put this solution on YOUR website!
.
The head gardener can mow the lawns in the city park twice as fast as his assistant.
Working together, they can complete the job in 1 1/3 hours.
How long would it take the head gardener working alone?
~~~~~~~~~~~~~~ 

According to the problem, the head gardener works as productive as two his assistants.


So, the problem statement is equivalent as if to say that three assistants will complete the job in 1 1/3 hours, 
or in 60 + 20 = 80 minutes, working together.


Hence, one assistant will complete the job in 80*3 = 240 minutes.


For the head gardener, it will take only half of this time, i.e. 240/2 = 120 minutes,
or 2 (two) hours.    


ANSWER.  The head gardener will complete the job in 2 hours, working alone.

Solved MENTALLY, using reasoning and without using equations.



Answer by josgarithmetic(39621)   (Show Source): You can put this solution on YOUR website!
Head Gardener, speed
Assistant speed,
The two together do the mowing work in hours.


-


----------time needed for just the assistant to do the work.
Time for the head gardener alone, hours.
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Tutor @ikleyn has shown one good way to solve the problem mentally, using logical reasoning and simple arithmetic, without algebra and equations.

Here is a different such path to the answer.

Since the head gardener works twice as fast as his assistant, when working together he does 2/3 of the job.

So in the 4/3 hours it takes them to do the job together, the head gardener does 2/3 of the job; that means the number of hours it would take him to do the job alone is (3/2)(4/3) = 2.

ANSWER: 2 hours.
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

A slightly different approach:

1/3 hr = (1/3)*60 = 60/3 = 20 min
1 & 1/3 hr = 1 hr + 1/3 hr
1 & 1/3 hr = 60 min + 20 min
1 & 1/3 hr = 80 min

Consider a lawn that is 900 sq ft. You can replace 900 with any multiple of 3.
gardener gets 2/3 of that, they get (2/3)*900 = 600 sq ft
assistant gets 1/3 of that, they get (1/3)*900 = 300 sq ft
The fractions add to 1, and 2/3 is twice that of 1/3

gardener's rate = (600 sq ft)/(80 min)
gardener's rate = 7.5 sq ft per min

Now compute the time needed when the gardener works alone
rate*time = amount done
time = (amount done)/(rate)
time = (900 sq ft)/(7.5 sq ft per min)
time = 120 min
time = 2 hours
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