Write an equation that using the variable t to represent the time it takes to paint the room together that can be used to solve the problem.

Question

Kristen can paint a room in 20 hours. Karissa can paint the same room in 12 hours. How long does it take for both Kristen and Karissa to paint the room if they are working together?

Write an equation that using the variable t to represent the time it takes to paint the room together that can be used to solve the problem.

Answer 1

Kristen paints the room in 20 hours

Karissa paints the room in 12 hours

Kristen paints 1/20 of the room in one hour

Kaeissa paints 1/12 of the room in one hour

t=# of hours it takes them to paint the room together

(1/20)(t)+(1/12)(t)=1 (job finished)

multiply both sides by the LCM of 20 and 12, which is 60

(60)(1/20)(t)+(60)(1/12)(t)=(60)(1)

3t+5t=60

8t=60

t=60/8

t=7 4/8=7 1/2 hours together

another simpler solution...

1/20+1/12=3/60+5/60=8/60 of the room painted together in one hour, or 8/60=2/15 of the room painted together in one hour

we want 15/15 of the room painted

how many hours will that take ?

2 hours, 4/15 done

3 hours, 6/15 done

4 hours, 8/15 done

5 hours, 10/15 done

6 hours, 12/15 done

7 hours, 14/15 done

7 1/2 hours 15/15 done

the simplest thing to do is 15/2=7 1/2

 

 

Answer 2

Focus on your units.  Both girls have a rate for room painting that is measured in rooms/hour.  This is no different than a car's speed being measured in miles/hour.

 

So if we multiply the rooms/hour by time (in hours), we could get how many total rooms we could paint.  For example, Kristen could paint R = (1 room/ 20 hour)*10 hours = 1/2 of the room in 10 hours.

 

We just need to combine both girls' rates in one equation.

 

Our equation would start with:

 

(1 room/20 hours)*t + (1 room/12 hours)*t = 1 room

 

The left side represents Kristen + Karissa, and the right side our desired output, the one room.  This we can solve for t.  Taking out the words, we end up with:

 

t/20 + t/12 = 1 , which is nothing more than a fraction problem.

 

3t/60 + 5t/60 = 1 (common denominator)

8t/60 = 1 (combing like terms)

8t = 60 (multiply both sides by 60)

t = 7.5 (divide both sides by 8)

 

So the time needed is 7.5 hours.  This makes sense, as it is faster than either girl working alone.