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Question 956559: a man does half as much work as his son in three - four of the time .if they take 18 days to complete the work .how many days will take to do work alone
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the general equation is rate * time = quantity of work.
the quantity of the work is equal to 1 job.
for the son, you get:
r * t = 1
solve for r to get r = 1/t
for the father you get:
r * 3/4 * t = 1/2
solve for r to get r = 4/6t which can be simplified to 2/3t
the ratio of the father's rate to the son's rate is 2/3t divided by 1/t.
this is the same as 2/3t * t which results in 2/3.
the father works at 2/3 the rate as the son.
when they work together, they can finish a job in 18 days.
you get:
t = 18
rate of the son = r
rate of the father = 2/3 * r
q is equal to 1.
when they work together, their rates are additive.
you get:
(r + 2/3 * r) * 18 = 1
use this equation to solve for r.
you get:
(3/3 * r + 2/3 * r) * 18 = 1
combine like terms to get:
5/3 * r * 18 = 1
simplify further to get:
(5 * 18)/3 * r = 1
simplify further to get:
30 * r = 1
solve for r to get r = 1/30.
that's the rate of the son.
the rate of the father is 2/3 * r which is equal to 2/90 which can be simplified to 1/45.
you have:
rate of the son is 1/30 of the job in 1 hour.
rate of the father is 1/45 of the job in 1 hour.
to confirm, replace these rates in the equation where both of them are working together to get:
(r + 2/3 * r) * 18 = 1
the equation becomes:
(1/30 + 1/45) * 18 = 1 which becomes:
(3/90 + 2/90) * 18 = 1 which becomes:
5/90 * 18 = 1 which becomes:
90/90 = 1 which becomes 1 = 1.
the rates are confirmed as good.
you now know the rates of the father and the son.
the question is:
how many days will it take to do the work alone.
the same equation is used, but now their rates are separate.
for the son:
r * t = q becomes 1/30 * t = 1 which gets you t = 30 days.
for the father:
r * t = q becomes 1/45 * t = 1 which gets you t = 45 days.
working together, they can do the job in 18 days.
working separately, the son can do the job in 30 days and the father can do the job in 45 days.
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