Question 257208
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.