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Question 909187: Computer encoder A can finish a job in as many days as encoder B. Encoder C can finish the same job in 2 days less time than encoder B. Together, the three encoders worked for 2 days; after which, encoder A and B got sick and were forced to stop working leaving encoder C alone to finish the job in three more days. Find the rate of each encoder.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = quantity.
quantity = 1 job
let x = number of days A can do the job alone.
let x = number of days B can do the job alone.
let (x-2) = number of days C can do the job alone.
the rate for A is 1/x of the job in one day.
the rate for B is 1/x of the job in one day.
the rate for C is 1/(x-2) of the job in one day.
this is determined by taking the equation of rate * time = quantity and solving for rate to get rate = quantity / time.
all 3 work for 2 days.
the formula for that is:
(1/x + 1/x + 1/(x-2) * 2 = part 1 of the job.
C finishes the rest of the job in 3 days.
the formula for that is:
(1/(x-2) * 3 = part 2 of the job.
simplify these equations to get:
2/x + 2/x + 2/(x-2) = part 1 of the job.
3/(x-2) = part 2 of the job.
since part 1 and part 2 of the job equals the whole job, you get:
2/x + 2/x + 2/(x-2) + 3/(x-2) = 1
multiply both sides of this equation by x * (x-2) to get:
2 * (x-2) + 2 * (x-2) + 2 * x + 3 * x = x * (x-2)
simplify this to get:
2x - 4 + 2x - 4 + 2x + 3x = x^2 - 2x
combine like terms to get:
9x - 8 = x^2 - 2x
subtract 9x from, and add 8x to, both sides of this equation to get:
0 = x^2 - 2x - 9x + 8
combine like terms to get:
0 = x^2 - 11x + 8
factor x^2 - 11x + 8 using the quadratic formula to get:
x = 10.21699057 or x = .783009434
x can't be .783009434 because x - 2 would then be negative and that isn't possible.
your solution must be x = 10.21699057 or not at all.
when x = 10.21699057, your rates becomes:
rate of A = 1/x = .0978761792
rate of B = 1/x = .0978761792
rate of C - 1/(x-2) = .1216990566
using these values, your equation of:
2/x + 2/x + 2/(x-1) + 3/(x-1) = 1 becomes:
2 * rate of A plus 2 * rate of B plus 2 * rate of C plus 3 * rate of C = 1
replace rate of A with .0978761792 and replace rate of B with .0978761792 and replace rate of C with .1216990566 and you get:
2 * .0978761792 + 2 * .0978761792 + 2 * .1216990566 + 3 * .1216990566 which can be simplified to:
4 * .0978761792 + 5 * .1216990566 = 1 which can be finally simplified to:
1 = 1
this confirms the solution is good.
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