# Math Help - Work rate word problem

1. ## Work 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?

2. 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

3. Originally Posted by Zerodollarbaby
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?
$Time(R_1 + R_2) = Job \ done$

$10(x) = 4$

$6(x+y) = 4$

$10x = 4x + 4y$

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

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

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

$x = 0,4$

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

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

$y = \frac{1}{15}$

4. Hello, Zerodollarbaby!

Here is yet another approach . . .

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 $\frac{1}{10}$ of the job.

Chris takes $x$ hours to paint the rooms.
. . In one hour, he completes $\frac{1}{x}$ of the job.

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

But we are told that together it takes them 6 hours to paint the rooms.
. . In one hour, they can complete $\frac{1}{6}$ of the job.

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

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

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