Question 488292: Please help me solve this word problem, I do not understand what to do.
It takes John 30 minutes to clean his house, When his wife helps him, it only takes him 18 minutes. How long would it take his wife by herself?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = units of work
the unit of work is equal to 1 which is the house.
john takes 30 minutes to clean the house.
if time is in minutes, then the equation for john is:
rate * 30 = 1
divide both sides of this equation by 30 and you get:
rate = 1/30
this means that john can clean 1/30 of the house in 1 minute.
when john's wife helps him, they both can clean the house in 18 minutes.
rate * time = units of work.
the unit of work is still the house so it is equal to 1
the rate of work is the combined rate of john and his wife.
the time is takes is now 18 minutes.
the formula of rate * time becomes:
(john's rate + john's wife's rate) * 18 = 1
we know john's rate because we just solved for it.
we'll make john's wife's rate = x
our formula becomes:
(1/30 + x) * 18 = 1
we multiply this out to get:
18/30 + 18x = 1
we subtract 18/30 from both sides of this equation to get:
18x = 1 - 18/30
we simplify this to get:
18x = 12/30
we divide both sides of this equation by 18 to get:
x = (12/30) / 18 = 1/45
this means that john's wife's rate is equal to 1/45.
john's rate is 1/30 of the house in one minute.
john's wife's rate is 1/45 of the house in one minute.
i believe we will see that it will take john's wife 45 minutes to clean the hours by herself.
the formula is rate * time = units of work.
the units of work is still equal to 1.
the wife's rate is equal to 1/45.
the formula becomes:
1/45 * time = 1
multiply both sides of this equation by 45 to get:
time = 45 minutes.
when they work together, their rates are additive.
the formula becomes:
(1/30 + 1/45) * time = 1
this simplifies to:
(3/90 + 2/90) * time = 1 which simplifies further to:
5/90 * time = 1 which simplifies further to:
1/18 * time = 1
multiply both sides of this equation by 18 to get:
time = 18
the number work out ok so we have the answer.
the answer is:
john's wife takes 45 minutes to clean the house by herself.
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