Question

Henry and Irene working together can wash all the windows of their house in 5 h 52 min. Working alone, it takes Henry
4
1
2
h more than Irene to do the job. How long does it take each person working alone to wash all the windows?

Answer 1

It takes Henry 4 1/2 hours more

Answer 2

It takes them 1hr 48 min=1.8 hours to do one job.
So they can do 1/1.8=5/9 jobs per hour.
So if Henry's rate alone is x jobs per hour, Irene's is (5/9-x) jobs per hour.
Thus, it takes 1/x hours for Henry to do 1 job, while it takes Irene 1/(5/9-x)=9/(5-9x) hours.
Now, it takes Henry 1.5 hours longer than Irene, so 1/x=9/(5-9x)+3/2

Solving this yields:

2(5-9x)=18x+3x(5-9x)
10-18x=18x+15x-27x^2
27x^2-51x+10=0
(9x-2)(3x-5)=0
x=2/9 or x=5/3.
We test these:

If Henry's rate is 2/9, then Irene's is 5/9-2/9=3/9
If his rate is 5/3, hers is -10/9.

Since the second solution is negative for Irene's rate, we throw it out, and that leaves us with:

Henry's rate=2/9 jobs/hour, so he can complete a job in 9/2=4.5 hours.
Irene's rate=3/9 jobs/hour, so she can complete a job in 9/3=3 hours.

(Not that the second solution is also strangely plausible. It just means that Irene dirties the windows at a rate of -10/9 per hour. Thus, Henry would do a job in 3/5 hours, while Irene at -9/10 hours. Thus, with Henry cleaning the windows, and Irene dirtying them at a slower rate, the job will still get done in 1.8 hours.

Answer 3

When I entered the answers it said they were incorrect

Answer 4

Let Henry and Irene be equal to the time it takes to wash all the windows in their house alone, respectively. Solve the following equations for Henry and Irene. Time has been converted to minutes for the equation calculations.\[\left\{\frac{1}{\text{Henry}}+\frac{1}{\text{Irene}}=\frac{1}{352},\text{ Henry}\text{ = }\text{Irene}+270\right\}\]\[\{\text{Henry}=864,\text{Irene}=594\} \]Henry = {Floor[864/60]hr,Mod[864,60]min} = {14 hr,24 min}
Irene = {Floor[594/60]hr,Mod[594,60]min} = {9 hr,54 min}