Lesson Using Fractions to solve word problems of the type WORKING TOGETHER

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Using fractions to solve word problems of the type "WORKING TOGETHER"

The purpose of this lesson is to present solutions of some word problems of the type "WORKING TOGETHER to complete a job".
The problems like "Painting a wall", "Painting a house", "Covering a roof", "Filling a reservoir", "Emptying a reservoir" etc. fall to this category.
In the lesson we consider those of this type of problems that can be solved using fractions.

Problem 1. Working together to cover a roof

Andrew can cover the roof of a house in 3 days.
Bill can make this job in 6 days.
How long will it take to Andrew and Bill to complete the job working together?

Solution
Since Andrew can cover the roof of a house in 3 days, he is covering 1%2F3

of the roof area per day.
Since Bill can complete this job in 6 days, he is covering 1%2F6

of the roof area per day.
So, working together, Andrew and Bill can cover 1%2F3+%2B+1%2F6

of the roof area per day.
Now you need to calculate this sum of two fractions, 1%2F3+%2B+1%2F6

. It requires standard operations:
    - to convert fractions to a common denominator, which is equal to 6 in this case;
    - then add numerators;
    - then reduce the resulting fraction:
1%2F3+%2B+1%2F6+=+2%2F6+%2B+1%2F6+=+%282%2B1%29%2F6+=+3%2F6+=+1%2F2

1%2F3+%2B+1%2F6+=+2%2F6+%2B+1%2F6+=+%282%2B1%29%2F6+=+3%2F6+=+1%2F2

.
Thus, working together, Andrew and Bill can cover 1%2F2

of the roof area per day.
Hence, it will take 2 days for Andrew and Bill to complete this job working together.

Answer. Working together, Andrew and Bill can complete the job in 2 days.

Problem 2. Installing solar panels

One team of workers can install solar panels on the roof of a house in 15 days by covering the entire roof area.
The second team of workers can complete this job in 10 days.
How long will it take for two teams to complete the job working together?

Solution
Since the first team can install solar panels on the roof of a house in 15 days by covering the entire roof area, the team is covering 1%2F15

of the roof area per day.
Since the second team can complete this job in 10 days, this team is covering 1%2F10

of the roof area per day.
Working together, two teams can cover 1%2F15+%2B+1%2F10

of the roof area per day.
Now you need to calculate this sum of two fractions, 1%2F15+%2B+1%2F10

. It requires standard operations:
    - to convert fractions to a common denominator, which is equal to 30 in this case;
    - then add numerators;
    - then reduce the resulting fraction:
1%2F15+%2B+1%2F10+=+2%2F30+%2B+3%2F30+=+%282%2B3%29%2F30+=+5%2F30+=+1%2F6

1%2F15+%2B+1%2F10+=+2%2F30+%2B+3%2F30+=+%282%2B3%29%2F30+=+5%2F30+=+1%2F6

.
Thus, working together, two teams can cover 1%2F6

of the roof area by solar panels each day.
Hence, it will take 6 days for two teams of workers to complete this job working together.

Answer. Working together, two teams of workers can complete the job in 10 days.

Problem 3. Filling a reservoir with the water

One tube can fill the reservoir with the water in 12 hours.
The second tube can fill the reservoir with the water in 36 hours, if works separately.
How long will it take to fill the reservoir, if two tubes work simultaneously?

Solution
Since the first tube can fill the reservoir with the water in 12 hours, it is filling 1%2F12

of the reservoir volume per hour.
Since the second tube can fill the reservoir with the water in 36 hours, it is filling 1%2F36

of the reservoir volume per hour.
Working simultaneously, two tubes fill 1%2F12+%2B+1%2F36+=+3%2F36+%2B+1%2F36+=+%283%2B1%29%2F36+=+4%2F36+=+1%2F9

of the reservoir volume per hour.
Hence, the reservoir will be filled in 9 hours, if both tubes work simultaneously.

Answer. It will take 9 hours to fill the reservoir, if two tubes work simultaneously.

Problem 4. Emptying a reservoir

An elephant can drink all the water from a reservoir in 4 days.
The rhinoceros can drink all the water from the same reservoir in 12 days, if drinks alone.
How long will it take to empty the reservoir, if the elephant and the rhinoceros drink together?

Solution
Since the elephant can drink all the water in 4 days, he drinks 1%2F4

of the reservoir volume per day.
Since the rhinoceros can drink all the water in 12 days, he drinks 1%2F12

of the reservoir volume per day.
Drinking together, two animals empty 1%2F4+%2B+1%2F12+=+3%2F12+%2B+1%2F12+=+%283%2B1%29%2F12+=+4%2F12+=+1%2F3

of the reservoir volume per day.
Hence, the reservoir will be emptied in 3 days, if both animals drink together.

Answer. The reservoir will be emptied in 3 days, if the elephant and the rhinoceros drink together.

Problem 5. Filling a reservoir

The tube can fill a reservoir with the water in 3 days.
An elephant can drink all the water from the reservoir in 4 days.
How long will it take to fill the reservoir, if the tube fills it and the elephant drinks the water at the same rates?

Solution
Since the tube can fill the reservoir with the water in 3 days, it is filling 1%2F3

of the reservoir volume per day.
Since the elephant can drink all the water of the full reservoir in 4 days, he is drinking 1%2F4

of the reservoir volume per day.
If the tube is filling 1%2F3

of the reservoir volume per day and the elephant is drinking 1%2F4

of the reservoir volume per day,
then the net balance is 1%2F3+-+1%2F4

of the reservoir volume per day.
Now you need to calculate this difference of two fractions, 1%2F3+-+1%2F4

. It requires standard operations:
    - to convert fractions to a common denominator, which is equal to 12 in this case;
    - then subtract numerators;
    - then reduce the resulting fraction:
1%2F3+-+1%2F4+=+4%2F12+-+3%2F12+=+%284-3%29%2F12+=+1%2F12

1%2F3+-+1%2F4+=+4%2F12+-+3%2F12+=+%284-3%29%2F12+=+1%2F12

.

So, if the tube is filling 1%2F3

of the reservoir volume per day and the elephant is drinking 1%2F4

of the reservoir volume per day,
then the volume of the water in the reservoir will increase in 1%2F12

part of the total volume at the end of each day.
Hence, the reservoir will be filled in 12 days.

Answer. The reservoir will be filled in 12 days.

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