Question

Jane can clean the living room in 3 hours, kai in 6 hours and dana in 8 hours. If they work together, in how many minutes can they clean the entire room?

Answer 1

`" 1 hour "36" minutes"`

Explanation:

Let the total amount of work (effort) required to clean the room be `W`

Let the work rate per hour for Jane be `w_j`
Let the work rate per hour for Kai be `w_k`
Let the work rate pr hour for Dana be `w_d`

Let the time they all worked together be `t`

Then when working on their own we have:

`w_jxx3" hours" =W color(white)("ddd") =>color(white)("ddd") w_j=W/3`

`w_kxx6" hours"=W color(white)("ddd") =>color(white)("ddd") w_k=W/6`

`w_d xx8" hours"=Wcolor(white)("ddd") =>color(white)("ddd") w_d=W/8`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For any given time the combined effort ( count of rooms) is:

`tw_j+tw_k+tw_d`

Factor out the `t`

`t(w_j+w_k+w_d)`

but we need `t` such that

`t(w_j+w_k+w_d)=W`

but `w_j=W/3; color(white)("d")w_k=W/6; color(white)("d")w_d=W/8` so by substitution

`t(w_j+w_k+w_d)=Wcolor(white)("ddd")->color(white)("ddd")t(W/3+W/6+W/8)=W`

`color(white)("ddddddddddddddddd.d")->color(white)("ddd")t((8W)/24+(4W)/24+(3W)/24)=W`

`color(white)("ddddddddddddddddddd")->color(white)("ddd") t(15W)/24=W`

`t=24/(15cancel(W))xxcancel(W)`

`t=24/15" hours "->" 1 hour "36" minutes"`

Answer 2

They together will clean the living room in `96` minutes.

Explanation:

In `1` hour Jane cleans `1/3` part of the room.

In `1` hour Kai cleans `1/6` part of the room.

In `1` hour Dana cleans `1/8` part of the room

They together clean `(1/3+1/6+1/8)= (8+4+3)/24=15/24` part

of the room in `1` hour. Therefore, they together will clean the

full living room in `1/(15/24)=24/15= 1.6` hour.

`1.6` hr `=1.6*60 = 96` minutes.

They together will clean the living room in `96` minutes [Ans]