SOLUTION: First I just hope I'm in the right section, I have been trying to figure out how to solve this problem but I just can't come up with the right formula. An office has an old co

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Question 252467: First I just hope I'm in the right section, I have been trying to figure out how to solve this problem but I just can't come up with the right formula.

An office has an old copying machine and a new one. Working together it takes both machines 6 hours to make all the copies of the annual financial report. Working alone the old machine takes 15 hours longer than the new machine. How long would it take the new machine to make all the copies working alone?
Found 3 solutions by drk, edjones, richwmiller:
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
This is a job / time problem. Here is the formula
[first person's job / first person's time] x (time together) + [second person's job / second person's time] x (time together) = total number of jobs. Using variables,
-----
+J1%2FT1+%2A+X+%2B+J2%2FT2+%2A+X+=+Tj
working together = 6 = X.
Old machine alone = 15 hours longer than new - -> T1 = T2 + 15.
New machine alone = T2 hours.
J1 = J2 = 1 for both machines; they can do that 1 job.
Tj = total number of jobs is 1; "financial report"
So we get
+%281+%2F+%2815%2BT2%29%29%2A%286%29+%2B+1%2FT2%2A%286%29+=+1
Multiply both sides by (T2)(15+T2) - your common denominator. we get
6T2+%2B+6%2815%2BT2%29+=+%28T2%29%5E2+%2B+15T2
Set everything = 0 and combine like terms. We get
%28T2%29%5E2+%2B3T2+-+90+=+0
Factor and find good values for T2.
T2 = 8.105.
Since the new machine was T2, it would take ~ 8.105 hours alone.

Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
Let n=time it takes for new machine to do the job.
Old does 1/(n+15) the job in 1 hr
together: 1/n + 1/(n+15)=1/6 (in one hour)
6(n+15)+6n=n(n+15) multiply each side by 6n(n+15) (LCM) to eliminate fractions.
6n+90+6n=n^2+15n
12n+90=n^2+15n
n^2+3n-90=0 subtract the left side from the right.
n=8.1 hrs. quadratic formula (below)
.
Ed
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-90+=+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=%283%29%5E2-4%2A1%2A-90=369.

Discriminant d=369 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+369+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+369+%29%29%2F2%5C1+=+8.10468635614927
x%5B2%5D+=+%28-%283%29-sqrt%28+369+%29%29%2F2%5C1+=+-11.1046863561493

Quadratic expression 1x%5E2%2B3x%2B-90 can be factored:
1x%5E2%2B3x%2B-90+=+1%28x-8.10468635614927%29%2A%28x--11.1046863561493%29
Again, the answer is: 8.10468635614927, -11.1046863561493. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-90+%29

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
I moved the problem to word problems rate of work.
What do we know?
old machine takes x+15
in one hour old machine takes 1/(x+15)
new machine takes x hours
in one hour the new machine takes 1/(x)
o=6 and n=6
together they take 6 hours
n/x+o/(x+15)=1
6/x+6/(x+15)=1
x=8.10= new machine alone
x+15=23.1 old machine alone