SOLUTION: X can do a certain job in 10 hours and Y can do it in 12 hours. After X and Y have been working together for 1 hour, Z joins them, and the three completed the job in 3 more hours,

Question 992764: X can do a certain job in 10 hours and Y can do it in 12 hours. After X and Y have been working together for 1 hour, Z joins them, and the three completed the job in 3 more hours, How long will it take Z to do the job alone?
Found 2 solutions by ptaylor, Theo:
Answer by ptaylor(2198) (Show Source): You can put this solution on YOUR website!
Let t = time it takes z to do the job working alone. Then z works at the rate of 1/t of the job per hour
x works at the rate of 1/10 of the job per hr
y works at the rate of 1/12 of the job per hour
Together x and y works at the rate of 1/10 + 1/12 =12/120 + 10/120=22/120=11/60 of the job per hour
After 1 hour, together they complete (11/60)*1=11/60 of the job leaving 49/60 of the job yet to be done
Soooo, (11/60)*3 + (1/t)*3= 49/60, or
33/60 + 3/t=49/60. multiply each term by 60t
33t + 180=49t
16t=180
t=11.25 hrs
CK 33/60 + 3/11.25 =49/60
99/180 + 48/180 = 147/180
147/180=174/180
Hope this helps ----ptaylor
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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.

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