1. Working Alone

James, by himself, can paint 4 rooms in 10 hours. If he hires June to help, they can do the same job together in 6 hours. If he lets June work alone, how long will it take her to paint 4 rooms?

2. Originally Posted by symmetry
James, by himself, can paint 4 rooms in 10 hours. If he hires June to help, they can do the same job together in 6 hours. If he lets June work alone, how long will it take her to paint 4 rooms?
The general formula is,
$\frac{ab}{a+b}$
Where $a,b$ is the amount of time it takes alone and the fraction is the time together they work.

Thus, the Man can do it in $a=10$. The Woman can do it in $b$ hours. Together they spend,
$\frac{10b}{10+b}=6$
Multiply by denominator,
$10b=6(10+b)$
$10b=60+6b$
$10b-6b=60$
$4b=60$
$b=15$ hours.

3. ok

I NEVER knew there was such an equation for solving "working alone" problems.

I learned something new today.

Thanks a million!

4. Originally Posted by symmetry
I NEVER knew that was such an equation for solving "working alone" problems.
It is my own little formula that becomes useful at times.

Thanks a million!
No you do not need to pay me a million.
You can pay me a 1 dollar today
1/2 a dollar tomorrow
1/3 a dollar the next day
...

That would make me happy.

5. Originally Posted by symmetry
James, by himself, can paint 4 rooms in 10 hours. If he hires June to help, they can do the same job together in 6 hours. If he lets June work alone, how long will it take her to paint 4 rooms?
Another way to look at this problem:

We know that James can paint 4 rooms in 10 hours; thus, in 30 hours he can paint 12 rooms.

We know that together they can paint 4 rooms in 6 hours. Thus, in 30 hours, they can point 20 rooms.

From this, we can see that June can paint 8 rooms in 30 hours. Thus, in order to paint just 4 rooms, it would take 15 hours.

6. Originally Posted by symmetry
James, by himself, can paint 4 rooms in 10 hours. If he hires June to help, they can do the same job together in 6 hours. If he lets June work alone, how long will it take her to paint 4 rooms?
James paints 4/10=0.4 rooms per hour
suppose June paints x rooms per hour.

Together they paint x+0.4 rooms per hour.

They paint 4 rooms in 6 hours so their combined rate is 4/6=2/3 rooms per hour

So x+0.4=0.66.., so x=2/3-2/5=4/15 rooms per hour.

Time for June to paint 4 rooms = 4/x =4*15/4=15 hours.

RonL