Question 992764
rate of x is equal to a
rate of y is equal to b
rate of z is equal to c


rate * time = quantity


the total quantity required to complete is 1 job.
partial quantities are just shown as q until they are solved for.
you'll see how that works below.


x can do the job in 10 hours.


for x, rate * time = quantity becomes a * 10 = 1
solve for a to get:
a = 1/10
that's the rate that x works at.
quantity is 1 because we are looking for the rate that x works at to complete the whole job working alone.


for y, rate * time = quantity becomes b * 12 = 1
solve for b to get:
b = 1/12
that's the rate that b works at.
quantity is 1 because we are looking for the rate that y works at to complete the whole job working alone.


x and y work on the job for 1 hour.
their rates are additive.
rate * time = quantity
(a + b) * t = q
a is the rate that x works at which is equal to 1/10
b is the rate that y works at which is equal to 1/12
t is 1 hour
q is unknown
formula becomes:
(1/10 + 1/12) * 1 = q
here we are looking for how much of the job is completed by x and y working together without z.
that's why the quantity is shown as q, rather than 1.
solve for q and you will get that q = 11/60 of the job.
x and y working together for 1 hour will complete 11/60 of the job in that hour.


what remains of the job to still be done is 1 - 11/60 = 60/60 - 11/60 = 49/60.


z joins them and all three finish the job in 3 additional hours.


since their rates when working together are additive, the formula to use would be:


(a + b + c) * t = q.


all three working together for 3 hours will complete 49/60 of the job.


we have:
a = 1/10
b = 1/12
c = c because we don't know what it is yet.
t = 3 hours
q = 49/60


formula becomes:


(1/10 + 1/12 + c) * 3 = 49/60


combine 1/10 + 1/12 and you get 22/120 which simplifies to 11/60.


formula becomes:


(11/60 + c) * 3 = 49/60


simplify the left side of the equation to get:


33/60 + 3 * c = 49/60


subtract 33/60 from both sides of the equation to get:


3 * c = 16/60


divide both sides of the equation by 3 to get:


c = 16/180 which simplifies to 8/90.


the rate of z is 8/90 of the job in 1 hour.


check your original equation to see if this is correct.


(1/10 + 1/12 + 8/90) * 3 = 49/60


do the math to get:


49/60 = 49/60
this confirms the solution is correct.


you now know the rate of z which is equal to 8/90


working alone, the rate * time = distance formula becomes:


8/90 * t = 1


8/90 is the rate of z
t is the time that we want to solve for
1 is the portion of the job to be completed which is the whole job.


solve for t to get t = 1 / (8/90) = 1 * (90/8) = 11.25 hours.


z can do the job alone in 11.25 hours.


that's your solution.