SOLUTION: One group of workers can do a job in 8days. After this group has worked 3 days, another group joins it and together they complete the job in 3 more days. In what time could the sec

Question 551376: One group of workers can do a job in 8days. After this group has worked 3 days, another group joins it and together they complete the job in 3 more days. In what time could the second group have done the job alone? Please, thank you~ :)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
group 1 can do the job in 8 days.
this group works for 3 days.
then a second group joins it and they complete the job in 3 more days.
the formula is rate * time = units
the number of unis is equal to 1 (the job).
the rate of the first group is 1/8 of the job per day because they can complete the whole job in 8 days.
the rate of the second group is 1/x of the job per day because they can complete the whole job in x days (x is what we're trying to find).
the first group finishes 3/8 of the job in 3 days.
this means that 5/8 of the job remains to be done.
when they work together, the rates are additive, so we get:
(1/8 + 1/x) * 3 = 5/8
this is because it takes them 3 more days to finish 5/8 of the job.
simpllify this equation to get:
3/8 + 3/x = 5/8
multiply both sides of this equation by 8*x to get:
3*8*x/8 + 3*8*x/x = 5*8*x/8
simplify this to get:
3*x + 3*8 = 5*x
subtract 3*x from both sides of this equation to get:
3*8 = 5*x - 3*x
simplify to get:
24 = 2*x
divide both sides of this equation by 2 to get:
x = 12
it appears that this second group could complete the job by itself in 12 days.
let's see if that holds true.
since they can complete the job in 12 days, this means they can complete 1/12 of the job in 1 day.
that's their rate.
we go back to the original equation.
the first group completes 3/8 of the job in 3 days.
5/8 of the job remains.
they work with the second group to complete the remaining 5/8 of the job in 3 days.
their combined rates are 1/8 and 1/12.
our equation becomes:
(1/8 + 1/12) * 3 = 5/8
we simplify to get:
3/8 + 3/12 = 5/8
since 2/8 is equivalent to 9/24 and 3/12 is equivalent to 6/24 and 5/8 is equivalent to 15/24, our equation becomes:
9/24 + 6/24 = 15/24 which simplifies to:
15/24 = 15/24
this confirms the value of x is good and the fact that the second group, working alone, would be able to complete the job in 12 days.

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