Two friends are painting a living room. Ken can paint it in 6 hours working alone. If Barbie works alone, it will take her 8 hours. How long will it take working together?

Let,the total work is of `x` amount.

So,ken does `x` amount of work in `6 hrs`

So,in `1 hr` he will do `x/6` amount of work.

Now,Barbie does `x` amount of work in `8 hrs`

So,in `1 hr` she does `x/8` amount of the work.

Let,after working `t hrs` together the work will be finished.

So, in `t hrs` Ken does `(xt)/6` amount of work and Barbie does `(xt)/8` amount of work.

Clearly, `(xt)/6 + (xt)/8 =x`

Or, `t/6 + t/8 =1`

So, `t=3.43 hrs`

People work at different rates. So the time taken by different people to complete a set amount of work will also be different. This is what we need to model

Let the total amount of work required to complete the task be `W`

Let Ken's work rate per hour be `w_k`
Let Barbie's work rate per hour be `w_b`
Let the total time working together be `t`

If Ken works on his own he can complete the whole task in 6 yours

`"work rate "xx" time = work done.".............Equation(1)`

`color(white)("ddd")w_kcolor(white)("dddd")xxcolor(white)("ddd")6color(white)("d")=color(white)("ddd")W`

So `w_k=W/6 ...........................Equation(2)`

If Barbie works on her own she can complete the whole task in 8 hours

Using the above method

`w_b=W/8........................Equation(3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider `Eqn(1)` but combine the two work rates `Eqn(2)+Eqn(3)`

`color(white)("d") (w_bxxt) color(white)("d")+ color(white)("d")(w_kxxt)=W`

`(W/8xxt)+(W/6xxt)=W`

Factor out the `t`

`t(W/8+W/6)=W`

`t((3W)/24+(4W)/24)=W`

`t(7W)/24=W`

`t=(24cancel(W))/(7cancel(W))`

`t=24/7" hours"`

`t=3 3/7" hours"larr` Exact value

`t=3" hours and "(3/7xx60)`

`t= 3" hours and "25 5/7" minutes"larr" Exact value"`

First find out what fraction of the task they would each complete in `1` hour.

Ken will finish `1/6` of the task in `1` hour.

Barbie will finish `1/8` of the task in `1` hour.

If they work together, in one hour they will finish:

`1/6 +1/8` of the painting task.

`= (4+3)/24 = 7/24` is the fraction completed in one hour.

So to complete the whole task `(24/24)` we need to divide:

`24/24 div 7/24`

`=24/24 xx24/7`

`=24/7` hours.

This simplifies to `3 3/7` hours

Which is easier given as `3` hours and `3/7 xx60` minutes

`=3 " hours " and 25 5/7` minutes

or `3 " hours " and 26` minutes