Question 488292
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.