Question 206942
let x = rate of first technician
let y = rate of second technician.
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working together they can complete a mailing in 3 hours.
(x + y) * 3 = 1
(x + y) is their combined rate.
3 = number of hours
1 = units produced.
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they produced 1 unit of mailing in 3 hours working together.
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let the first technician be the faster mailer.
the first technician works at a rate of x mailings per hour.
the second technician works at a rate of y mailings per hour. 
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alone, the first technician can complete the mailing 2 hours faster than the other.
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let h = amount of time in hours it takes for the first technician to complete the mailing.
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x * h = 1 mailing
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since the first technician can complete the mailing in 2 hours less than the second technician, then (h+2) represents the time it takes the second technician to complete the mailing.
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y * (h+2) = 1 mailing
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since x * h = 1, then x = (1/h) which looks like {{{(1/h)}}}
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since y * (h+2) = 1, then y = (1/(h+2)) which looks like {{{(1/(h+2))}}}
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formula of (x+y)*3 = 1 becomes ((1/h) + (1/(h+2))) * 3 = 1
this looks like {{{((1/h) + (1/(h+2))) * 3 = 1}}}
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multiply both sides of this equation by (h)*(h+2) to get ((h+2) + (h)) * 3 = (h)*(h+2)
this looks like {{{((h+2) + (h)) * 3 = (h)*(h+2)}}}
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this becomes 6h + 6 = h^2 + 2h which looks like {{{6h + 6 = h^2 + 2h}}}
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this eventually results in h^2 - 4h - 6 = 0 which looks like {{{h^2 - 4h - 6 = 0}}}
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using the quadratic formula, the roots are either:
h = 5.16227766 or h = -1.16227766
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since h can't be negative, the only possible answer is h = 5.16227766
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if h = 5.16227766, then x = 1/5.16227766 = .193712943 and y = 1/7.16227766 = .13962039
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to test this out, we substitute for x and h in the equations we created that contain h in them.
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first equation is {{{((1/h) + (1/(h+2))) * 3 = 1}}}
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this becomes {{{((1/5.16227766) + (1/7.16227766)) * 3 = 1}}}
which results in 1 = 1 proving the value of h is correct.
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second equation is x * h = 1.
x = .193712943
h = 5.16227766
x * h = 1 proving the values for x and h are good.
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third equation is y * (h+2) = 1.
y = .13962039
h + 2 = 7.16227766
y * (h+2) = 1 proving the values for y and (h+2) are good.
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we started off with {{{(x+y) = 1}}}
we then got {{{(x*h) = 1}}}
we then got {{{(y*(h+2)) = 1}}}
we then got {{{x = (1/h)}}}
we then got {{{y = (1/(h+2))}}}
we then substituted in {{{(x+y) = 1}}} to get {{{((1/h) + (1/(h+2))) * 3 = 1}}}
we then solved for h.
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once we solved for h we were then able to complete the problem.
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