SOLUTION: Ronnie takes 4 hours longer than Tim to paint a room. if they work together they can paint the room in 9 hours. How long would it take each person if they worked alone? using fract

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Ronnie takes 4 hours longer than Tim to paint a room. if they work together they can paint the room in 9 hours. How long would it take each person if they worked alone? using fract      Log On


   

Question 883050: Ronnie takes 4 hours longer than Tim to paint a room. if they work together they can paint the room in 9 hours. How long would it take each person if they worked alone? using fractions not decimals in answer.
let a = Tim (X)
b= Ronnie (X+4)
t= 9 hrs
t/a +t/b=1
9/x+9/x+4=9
i feel this is the correct format, but im lost at this point
Found 3 solutions by josgarithmetic, josmiceli, algebrapro18:
Answer by josgarithmetic(39838) About Me  (Show Source):
You can put this solution on YOUR website!
Uniform Rates for doing work or a task or a job:
RT=J Rate, Time, Job.
The rate in this form is in amount of job per time. The numerator of the rate is quantity of job and the denominator in the rate is quantity of time. You also should understand that for a given agent doing some work, the ratio J/T is a constant, which is R for the agent.

Ronnie takes 4 hours longer than Tim to paint a room.
Let x = time needed for Tim to do the job.
Tim's rate is 1%2Fx.
Ronnie's rate is 1%2F%28x%2B4%29.

if they work together they can paint the room in 9 hours.
Their combined rate is 1%2F9.

The sum of their rates individually is the rate for the two working together or simultaneously:
highlight%281%2Fx%2B1%2F%28x%2B4%29=1%2F9%29.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +t+ = time in hours it takes
Tim to paint the room
+t+%2B+4+ = time in hours for
Ronnie to paint the room
-----------------------
Add their rates of working to get
their rate working together
+1%2Ft+%2B+1%2F%28+t%2B4%29+=+1%2F9+
Multiply both sides by +9%2At%2A%28+t%2B4+%29+
+9%2A%28+t%2B4+%29+%2B+9t+=+t%2A%28+t%2B4+%29+
+9t+%2B+36+%2B+9t+=+t%5E2+%2B+4t+
+t%5E2+-+14t+=+36+
Complete the square
+t%5E2+-+14t+%2B+%2814%2F2%29%5E2+=+36+%2B+%2814%2F2%29%5E2+
+t%5E2+-+14t+%2B+49+=+36+%2B+49+
+%28+t+-+7+%29%5E2+=+85+
+t+-+7+=+sqrt%2885%29+
+t+=+7+%2B+sqrt%2885%29+ ( Tim's time )
+t+%2B+4+=+11+%2B+sqrt%2885%29+ ( Ronnie's time )
------------------------
check:
+1%2Ft+%2B+1%2F%28+t%2B4%29+=+1%2F9+
+1%2F%28+7+%2B+sqrt%2885%29%29+%2B+1%2F%28+11+%2B+sqrt%2885%29%29+=+1%2F9+

+99+%2B+9%2Asqrt%2885%29+%2B+63+%2B+9%2Asqrt%2885%29+=+77+%2B+18%2Asqrt%2885%29+%2B+85+
+99+%2B+63+=+77+%2B+85+
+162+=+162+
OK
Hope I got it



Answer by algebrapro18(249) About Me  (Show Source):
You can put this solution on YOUR website!
Well you had the right idea but your set up is a little flawed.
If we let R = the time it takes Ronnie to paint a room and T = the time it takes Tim to pain a room then we have the following equations:
R = 4+T
R+T = 9
Now we see that R is already by its self in the first equation so lets just plug that in for R in the second equation and solve for T.
R+T = 9
4+T+T = 9
4+2T = 9
2T = 5
T = 5/2
So we know that it takes Tim 5/2 hours to paint the room which is 2 and a half hours. Now we can plug that 5/2 into the first equation and find how long it takes Ronnie.
R = 4+T
R = 4 + 5/2
R = 8/2 + 5/2
R = 13/2
So it takes Tim 5/2 hours or 2 and a half hours to paint the room while it takes Ronnie 13/2 hours or 6 and a half hours to paint the same room.