|
Question 817746: Jugi can paint a room in 5 weeks. Mac can paint same room in 7 weeks. If they work together for 2 weeks then Jugi stops, how long can Mac finish it?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! jugi can paint the room in 5 weeks.
this means that jugi can paint the room at a rate of 1/5 of the room per week.
mac can paint the same room in 7 weeks.
this means that mac can paint the same room at a rate of 1/7 of the room per week.
they work together for 2 weeks and then mac finishes the room.
working together means their rates are additive.
this means that jugi's rate plus mac's rate will be equal to 1/5 + 1/7.
add these 2 rates together as follows:
1/5 * 7/7 = 7/35
1/7 * 5/5 = 5/35
their rates now have a common denominator and they can be added together.
you get 7/35 + 5/35 = 12/35
that's their rate working together.
for 2 weeks they are working at the combined rate of 12/35 of the room per week.
this means that, after 2 weeks, they have completed 2 * 12/35 of the room which means they have completed 24/35 of the room.
that means that they have 35 - 24 = 11/35 of the room left to complete.
this is because 11/35 plus 24/35 = 1 which represents the whole room.
so mac is left alone and mac works at the rate of 1/7 of the room per week.
the formula used is R * T = Q
R = rate
T = time
Q = quantity
The rate is equal to 1/7 of the room per week which is the rate that mac paints the room.
the time is what we want to find.
the quantity is the 11/35 of the room that still needs to be painted.
the formula becomes:
1/7 * T = 11/35
multiply both sides of this equation by 7/1 and you get:
7/1 * 1/7 * T = 7/1 * 11/35
simplify to get:
T = 77/35
it will take mac an addition 77/35 weeks to finish the room.
let's see how that works out.
they work together for 2 weeks.
mac works at the rate of 1/7 of the room per week.
jugi works at the rate of 1/5 of the room per week.
in the 2 weeks:
mac has finished 2/7 of the room.
jugi has finished 2/5 of the room.
to add these 2 fractions together you need to find the common denominator.
the common denominator is 35.
2/7 is equivalent to 10/35
2/5 is equivalent to 14/35
add 10/35 and 14/35 and you will get 24/35 which is the fraction of the room that is completed in 2 weeks.
now we get to mac finishing the rest of the room.
he works at the rate of 1/7 of the room per week and he works for 77/35 weeks.
the formula is still R * T = Q
R = 1/7
T = 77/35
formula becomes 1/7 * 77/35 = Q
we solve for Q to get Q = 1/7 * 77/35 = 11/35
jugi and mac, working together finished 24/35 of the room.
mac working alone finished 11/35 of the room.
24/35 added to 11/35 equals 35/35 of the room which is equal to 1 which means the whole room has been painted.
it took mac 77/35 weeks to finish the room by himself.
2 weeks + 77/35 weeks is equal to 70/35 + 77/35 which is equal to 147/35 weeks.
convert that into a mixed fraction and you get 4 and 7/35 weeks which can be further simplified to 4 and 1/5 weeks.
the whole problem involves the use of the formula of R * T = Q
R = rate of work
T = hours or weeks or whatever represents the amount of time to do the work.
Q represents the quantity consumed or produced.
in this case the quantity produced is equal to 1 painted room.
when they work together, their rates are additive.
|
|
|
| |
