Answer:
25.7 minutes
Round to 26 minutes if they ask for the nearest minute.
Explanation:
Ah, the old rate-of-work problem.
Let me ask you this, just to check for understanding:
If it took the mother and the daughter EACH 45 minutes to clean the house (instead of the daughter being slower), how long would it take? Two people working at the same rate can do the job in half the time, right? So 22.5 minutes, half of the 45 minutes it takes one person.
Now, the daughter is slower, but not THAT much slower, so our answer should come out to be more than 22.5 minutes, but not a whole lot more. Does that all make sense? We'll use it later to see if our answer makes sense.
So let's figure out how much of the "job" (cleaning the house) each can do in 1 minute:
Mom: 1 job in 45 minutes = 1/45 job/minute. She can clean one forty-fifth of the house every minute. Make sense?
But daughter is slower: 1 job in 60 minutes = 1/60 job/minute. She can clean only one sixtieth of the house every minute (because she's slower, right?).
So now we need to let them work together at their different rates, but for the same amount of time, which we'll call t. The full/whole/one house needs to be clean at the end of time t:
(Mom's rate x t) + (daughter's rate x t) = 1 job done
(1/45 x t) + (1/60 x t) = 1
So we need to solve for t, and to do so we need to find a common denominator for 45 and 60. ANY common denominator will work, but the LEAST common denominator (LCD) is 180.
So convert 1/45 to 4/180.
And convert 1/60 to 3/180.
Now our equation is: 4/180t + 3/180t = 1
Add the fractions to get: 7/180t =1
Solve for t to get: t = 180/7 = 25.7 minutes