SOLUTION: This weekend, while my fiance and daughter were out of town, I decided to paint my living room. I am a horrible painter, but my mom Mary, is excellent. It takes me 4 more hours t

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: This weekend, while my fiance and daughter were out of town, I decided to paint my living room. I am a horrible painter, but my mom Mary, is excellent. It takes me 4 more hours t      Log On


   

Question 206336: This weekend, while my fiance and daughter were out of town, I decided to paint my living room. I am a horrible painter, but my mom Mary, is excellent. It takes me 4 more hours to paint room the same size of my living room than it takes my mom. We painted the room in 5 hours together. If we worked independently, how long would it take to finish painting the room?
Found 2 solutions by Alan3354, ankor@dixie-net.com:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
It takes me 4 more hours to paint room the same size of my living room than it takes my mom. We painted the room in 5 hours together. If we worked independently, how long would it take to finish painting the room?
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It takes your mom x hours to paint the room.
It takes you x + 4 hours.
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Your mom does 1/x of the job per hour.
You do 1/(x+4) per hour.
Together, you do 1/x + 1/(x+4)
1/x + 1/(x+4) = ((x+4) + x)/x(x+4)
That's the part of the job you do together per hour.
Hours/job is the inverse
= x(x+4)/(2x+4) = 5 hours
x^2+4x = 10x+20
x^2 - 6x - 20 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-6x%2B-20+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-6%29%5E2-4%2A1%2A-20=116.

Discriminant d=116 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+116+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+116+%29%29%2F2%5C1+=+8.3851648071345
x%5B2%5D+=+%28-%28-6%29-sqrt%28+116+%29%29%2F2%5C1+=+-2.3851648071345

Quadratic expression 1x%5E2%2B-6x%2B-20 can be factored:
1x%5E2%2B-6x%2B-20+=+%28x-8.3851648071345%29%2A%28x--2.3851648071345%29
Again, the answer is: 8.3851648071345, -2.3851648071345. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B-20+%29

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Ignore the negative answer.
x = 3+ sqrt(29) hours for your mom
x+4 = 7 + sqrt(29) for you


Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
It takes me 4 more hours to paint room the same size of my living room than it takes my mom.
We painted the room in 5 hours together.
If we worked independently, how long would it take to finish painting the room?
:
Let t = time required by mom
then
(t+4) = time required by me
;
Let the completed job = 1
:
5%2Ft + 5%2F%28%28t%2B4%29%29 = 1
:
Multiply equation by t(t+4), results
5(t+4) + 5t = t(t+4)
:
5t + 20 + 5t = t^2 + 4t
:
Arrange as a quadratic on the right:
0 = t^2 + 4t - 10t - 20
:
t^2 - 6t - 20 = 0
:
Use the quadratic equation to find t
:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
in the problem:
x=t: a=1; b=-6; c=-20
t+=+%28-%28-6%29+%2B-+sqrt%28-6%5E2+-+4+%2A+1+%2A+-20+%29%29%2F%282%2A1%29+
t+=+%286+%2B-+sqrt%2836+-+%28-80%29+%29%29%2F2+
t+=+%286+%2B-+sqrt%2836+%2B+80+%29%29%2F2+
t+=+%286+%2B-+sqrt%28116+%29%29%2F2+
the positive solution
t+=+%286+%2B+10.77+%29%29%2F2+
t = 16.77%2F2
t = 8.385 hr is Mom's time alone
then
8.385 + 4 = 12.385 is my time alone
:
:
Check solution:
5%2F8.385 + 5%2F12.385 =
.596 + .404 = 1