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Question 1035896: Jack and Jill finished a job together in 15 days. But when Jill joined Jack only after he worked for 12 days, they finished the same job in 6 days more. How many days can each of them finish the job alone?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = quantity.
let jack's rate of work = a.
let jill's rate of work = b.
when they work together, they finish the job in 15 days.
rate * time = quantity becomes (a + b) * 15 = 1.
when jack works alone for 12 days and then jill joins him for 6, the formula becomes:
a * 12 + (a + b) * 6 = 1.
you have two equations that need to be solved simultaneously.
they are:
(a + b) * 15 = 1
a * 12 + (a + b) * 6 = 1
simplify both equations to get:
15a + 15b = 1
18a + 6b = 1
we'll solve these by elimination.
multiply both sides of the first equation by 2 and multiply both sides of the second equation by 5 to get:
30a + 30b = 2
90a + 30b = 5
subtract the first equation from the second equation to get:
60a = 3
divide both sides of this equation by 60 to get:
a = 3/60 = 1/20.
the original equations you started with are:
(a + b) * 15 = 1
a * 12 + (a + b) * 6 = 1
go back to the first of these equations and replace a with 1/20 to solve for b.
start with:
(a+b) * 15 = 1
replace a with 1/20 to get:
(1/20 + b) * 15 = 1
simplify to get:
15/20 + 15b = 1
subtract 15/20 from both sides of this equation to get:
15b = 5/20
divide both sides of this equation by 15 to get:
b = 5/(20*15) = 1/(20*3) = 1/60.
you have a = 1/20 and b = 1/60
a is the rate that jack works.
b is the rate that jill works.
so you have:
jack's rate of work is 1/20 of the job in 1 day.
jill's rate of work is 1/60 of the job in 1 day.
you can confirm this is true by going back to the original equations to make sure they are both true when you are using these values for a and b.
the original equations are:
(a + b) * 15 = 1
a * 12 + (a + b) * 6 = 1
replace a with 1/20 and b with 1/60 to get:
(1/20 + 1/60) * 15 = 1
1/20 * 12 + (1/20 + 1/60) * 6 = 1
simplify these equations to get:
4/60 * 15 = 1
36/60 + 18/60 + 6/60 = 1
simplify further to get:
60/60 = 1
60/60 = 1
both equations are true, confirming the solutions are correct.
you were asked to find how long each would take working alone.
for jack, the rate * time formula becomes 1/20 * t = 1
solve for t to get that jack can do the job in 20 days working alone.
for jill, the rate * time formula becomes 1/60 * t = 1
solve for t to get that jill can do the job in 60 days working alone.
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