Ray by himself can paint four rooms in 10 hours. If he hires Chris to help, together they can do the same job in 6 hours. If Ray lets Chris work alone, how long will it take for Chris to paint the four rooms?

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- Apr 5th 2005, 03:52 PMZerodollarbabyWork rate word problem
Ray by himself can paint four rooms in 10 hours. If he hires Chris to help, together they can do the same job in 6 hours. If Ray lets Chris work alone, how long will it take for Chris to paint the four rooms?

- Apr 5th 2005, 05:09 PMMathGuru
Welcome to MHF zerodollarbaby,

First lets figure out how fast ray works in rooms per hour.

Ray by himself can paint 4 rooms in 10 hours.

R = 4 rooms / 10 hours= 0.4 rooms/hour

Then lets figure out how fast ray and chris work together

R+C= 4 rooms/ 6 hours = .66' rooms/hour

Now lets subtract Ray's work rate from Ray and Chris' work rate

(R+C)-R=C

(.66')-(.4)=C

.266'=C

Chris paints at a rate of .266' rooms/hour

now divide the number of rooms by his work rate to find how many hours it would take him alone.

4 rooms / (.266' rooms/hour) ;rooms cancel and the unit becomes hours

Ans=15 hours - May 26th 2007, 05:48 AMjanvdl
$\displaystyle Time(R_1 + R_2) = Job \ done $

$\displaystyle 10(x) = 4 $

$\displaystyle 6(x+y) = 4 $

$\displaystyle 10x = 4x + 4y $

$\displaystyle y = \frac{4}{6} x $

$\displaystyle Set \ y's \ value \ into \ the \ equation $

$\displaystyle 6(x + \frac{4}{6} x) = 4 $

$\displaystyle x = 0,4 $

$\displaystyle Set \ x = 0,4 \ into \ equation $

$\displaystyle 6(0,4 + y) = 4 $

$\displaystyle y = \frac{1}{15} $ - May 26th 2007, 06:03 AMSoroban
Hello, Zerodollarbaby!

Here is yet another approach . . .

Quote:

Ray by himself can paint four rooms in 10 hours.

If he hires Chris to help, together they can do the same job in 6 hours.

If Ray lets Chris work alone, how long will it take for Chris to paint the four rooms?

Ray takes 10 hours to paint the rooms.

. . In one hour, he completes $\displaystyle \frac{1}{10}$ of the job.

Chris takes $\displaystyle x$ hours to paint the rooms.

. . In one hour, he completes $\displaystyle \frac{1}{x}$ of the job.

Together, in one hour, they can complete: .$\displaystyle \frac{1}{10} + \frac{1}{x}$ of the job.

But we are told that together it takes them 6 hours to paint the rooms.

. . In one hour, they can complete $\displaystyle \frac{1}{6}$ of the job.

There is our equation! . . . $\displaystyle \frac{1}{10} + \frac{1}{x}\:=\:\frac{1}{6}$

Multiply by $\displaystyle 30x:\;\;3x + 30 \:=\:5x\quad\Rightarrow\quad 30 = 2x\quad\Rightarrow\quad x = 15$

Therefore, it will take Chris 15 hours to paint the rooms working alone.