Question 171319
if i understand this problem correctly.
sam is working on one machine.
bob is working on another machine.
i presume you are wanting to solve for x.
-----
rate * time = units produced
this would be the basic equation.
the number of units in this case is 1 machine for bob and 1 machine for sam.
-----
sam's rate is 1/x.
since rate * time = units produced, then
(1/x)*t = 1
dividing both sides of the equation by 1/x, we get:
t = 1/(1/x) = x
this means that sam takes x hours to complete 1 job.
-----
bob's rate is 1/(x+5)
since rate * time = units produce, then
(1/(x+5) * t = 1
dividing both sides of the equation by 1/(x+5), we get:
t = 1 / (1/(x+5) = x + 5
this means that bob takes x + 5 hours to complete 1 job.
-----
the problem states that both work for x hours.
this means that sam has finished his job since he only takes x hours to complete it.
this also means that bob, who is slower, still has 5 hours to go to finish his job.
sam, who is faster, only takes 3 hours to do what it takes bob 5 hours to do.
-----
what you have is a new job that takes 5 hours to complete at the rate of 1/(x+5), and that takes 3 hours to complete at a rate of 1/x.
-----
your equation is the same:
rate * time = units produced.
the units produced is still 1.
we have, however, a value for t.
-----
for bob to finish the job, the equation becomes:
rate = 1/(x+5)
time = 5
units produced = 1
r*t=u
(1/(x+5))*5=1
-----
for sam to finish the job, the equation becomes:
rate = 1/x
time = 3
units produced = 1
r*t=u
(1/x)*3 = 1
-----
since both of these equations = 1, then they are both = to each other.
so we have:
{{{(1/x)*3 = (1/(x+5))*5}}}
this is the same as:
{{{(3/x) = (5/(x+5))}}}
if we multiply both sides of the equation by (x+5), we get:
{{{(3*(x+5))/x = 5}}}
if we multiply both sides of the equation by x, we get:
{{{3*(x+5) = 5*x}}}
we can simplify this to be:
3*x + 15 = 5*x
15 = 2*x
x = 7.5
-----
if x = 7.5, then
x + 5 = 12.5
it takes sam 7.5 hours to do 1 job.
it takes bob 12.5 hours to do 1 job.
-----
sam's rate is 1/7.5 jobs per hour = .1333333..... jobs per hour.
bob's rate is 1/12.5 jobs per hour = .08 jobs per hour.
-----
to prove this is correct, we need to substitute in the original formulas.
the only formulas we have that we can use are:
bob takes 5 hours to complet a job that sam completes in 3 hours.
since rate * time = units
then
bob works 5 hours at a rate of (1/12.5) jobs per hour.
sam works 3 hours at a rate of (1/7.5) jobs per hour.
5 * (1/12.5) = .4
3 * (1/7.5) = .4
looks like they both did .4 of a job.
-----
this means that bob must have done .6 of his job when he quit.
let's see if that works out.
bob's rate is (1/12.5) jobs per hour.
he worked x hours.
x was found out to be 7.5
so 7.5 * (1/12.5) = .6
bob finished .6 of his job when he stopped working.
he would have taken 5 more hours to finish.
sam working on it instead and only took 3 hours.
-----
value of x is good.
equations prove out.
now all you need to do is understand what was done.
hopefully this helped.
-----
the key was rate times time = units produced.
units produced was 1 for each of them.
their respective rates were given:
1/x for sam
1/(x+5) for bob.
the key to solving the problem was the job that was left to finish.
we didn't know what proportion of bob's job was left, but we did know that we could look at the remaining portion as 1 job.
we discovered what proportion was left after we solved for x.
-----
any questions email to gonzo@gmx.us and i'll try to help you understand better if you are still confused.
-----