Question 257208: Working together, Kate and Jean can close the restaurant in 2/3 of an hour.
Working alone, Jean can do the job in 1 hour less time than Kate, when she works
alone. How long does it take Jean to do the job alone? Write your answer in
terms of hours.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! basic formula is r * t = u where:
r is the rate of units per hour.
t is the time in hours.
u is the number of units produced.
let k = rate that kate works at.
let j = rate that jean works at.
in this problem, the formula becomes:
(k+j)*(2/3) = 1 where:
k+j is the combined rate of kate and jean.
(2/3) is the number of hours it takes them to close the restaurant working together.
1 is the number of units produced. the unit is a closed restaurant and they closed one of them.
the problem states that jean can close the restaurant in 1 hours less than kate when they are both working alone.
if kate takes x hours to close the restaurant working alone, then we get:
k*x = 1
if jean takes 1 hour less, then we get:
j * (x-1) = 1
using these formulas, we can reduce the number of unknowns from 2 to 1 by making k and j equivalent to a formula in x.
k*x = 1 becomes k = 1/x when we divide both sides of that equation by x.
j*(x-1) = 1 becomes j = 1/(x-1) when we divide both sides of that equation by (x-1).
we get:
k = 1/x and j = 1/(x-1)
we go back to the formula of (k+j)*(3/2) = 1 and substitute for k and j to get:
(1/x + 1/(x-1)) * 2/3 = 1
the formula that had 2 unknown variables (k and j) now has only 1 (x).
we can solve this formula for x as follows:
multiply both sides of this equation by 3 to get:
(1/x + 1/(x-1)) * 2 = 3
multiply out the factors to get:
2/x + 2/(x-1) = 3
multiply both sides of the equation by x * (x-1) to get:
2 * (x-1) + 2*x = 3 * x * (x-1)
simplify to get:
2*x - 2 + 2*x = 3*x^2 - 3*x
combine like terms to get:
4*x - 2 = 3*x^2 - 3*x
subtract 4*x from both sides of the equation and add 2 to both sides of the equation to get:
0 = 3*x^2 - 3*x - 4*x + 2
combine like terms to get:
0 = 3*x^2 - 7*x + 2
this is the same as:
3*x^2 - 7*x + 2 = 0
factor this quadratic equation to get:
(3x-1) * (x-2) = 0
solve for x to get:
x = 1/3 or x = 2
plug these values into the original equations of k*x = 1 and j*(x-1) = 1 to see if they are valid.
when x = 1/3, j*(x-1) becomes j*(-2/3) which is negative and therefore invalid, so x = 1/3 is not good.
when x = 2, we get:
k*2 = 1 which leads to k = 1/2
j*(2-1) = 1 which leads to j*1 = 1 which leads to j = 1
plug these values for j and k into the original equation of (k+j)*(2/3) = 1 and you get:
(1/2 + 1)*(2/3) = 1 which becomes (3/2)*(2/3) = 1 which becomes 1 = 1 confirming the values for k and j are good.
we have k = 1/2 which means that kate can complete 1/2 of the job per hour.
we have j = 1 which means that jean can complete all of the job per hour.
we have x = 2.
this means that kate takes 2 hours to complete the job alone (x).
this means that jean takes 1 hour to complete the job alone (x-1).
k*x = 1 becomes 1/2 * 2 = 1 becomes 1 = 1 confirming kate's rate is good.
j*(x-1)= 1 becomes 1*(2-1) = 1 becomes 1*1 = 1 becomes 1 = 1 confirming jean's rate is good.
the question was:
How long does it take Jean to do the job alone? Write your answer in
terms of hours.
the answer is:
it takes jean 1 hour to do the job alone.
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