Question 997068
rate * time = quantity of work.


quantity of work = 1 job


jerry can lay a tile floor in 3 hours less time than jake.


let x = the amount of time that jake takes.
then x - 3 = the amount of time that jerry takes.


since rate * time = quantity, then:


for jake, rate * time = quantity becomes rate * x = 1


for jerry, rate * time = quantity becomes rate * (x-3) = 1


solve for rate in each of these equations and you get:


for jake, rate = 1/x


for jerry, rate = 1/(x-3)


when they work together, they can do the job in 2 hours.


their rates are additive when they work together.


time is equal to 2 hours.
quanity is still equal to 1 job.


rate * time = quantity becomes:


(1/x + 1/(x-3) * 2 = 1


you want to solve for x.


remove the parentheses to get:


2/x + 2/(x-3) = 1


multiply both sides of the equation by x * (x-3) to get:


(2 * x * (x-3) / x + (2 * x * (x-3)) / (x-3) = 1 * x * (x-3)


simplify to get:


2 * (x-3) + 2 * x = x * (x-3)


remove parentheses to get:


2 * x - 6 + 2 * x = x^2 - 3 * x


combine like terms to get:


4 * x - 6 = x^2 - 3 * x


subtract 4 * x from both sides of the equation and add 6 to both sides of the equation to get:


0 = x^2 - 7 * x + 6


solve for the factors of this quadratic equation to get:


(x - 6) * (x - 1) = 0


solve for x to get:


x = 6 or x = 1


x can't be equal to 1 because then x - 3 would be negative and negative values of time or rate are not allowed.


x has to be equal to 6 or not at all.


to confirm that x = 6 is good, go back to the original equation where they were both working together to get:


(1/x + 1/(x-3) * 2 = 1 becomes:


(1/6 + 1/(6-3)) * 2 = 1 which becomes:


(1/6 + 1/3) * 2 = 1 which becomes:


(1/6 + 2/6) * 2 = 1 which becomes:


3/6 * 2 = 1 which is equal to 1/2 * 2 = 1 which is equal to 1 = 1.


this confirms the value of x is good.


you are asked how long it would take jerry working alone.


rate * time = quantity


quantity = 1
rate = 1/(6-3) = 1/3


your equation becomes:


1/3 * time = 1


solve for time to get time = 3 hours.


it would take jerry 3 hours to do the job working alone.