# Work word problem: how long for each to wash windows?

#### crazylum

##### New member
My textbook gives the following word problem:

Henry and Irene working together can wash all the windows of their house in 1h 48m. Working alone, it takes Henry 1.5h more than Irene to do the job. How long does it take each person working alone to wash all the windows?

I know the answer is 3 and 4.5, but I don't know how they arrived at this. Here is what I do know and have done:

1h 48m = 1.8h

Irene's work time: x
Henry's work time: x + 1.5
Working together: 1/x + 1/(x+1.5) = 1/1.8

I've multiplied through by 1.8x(x+1.5) to eliminate the denominators and ended up with the following:

1.8(x+1.5) + 1.8x = x(x+1.5)
1.8x + 2.7 + 1.8x = x[sup:32ih4xpe]2[/sup:32ih4xpe]+1.5x
3.6x +2.7 = x[sup:32ih4xpe]2[/sup:32ih4xpe]+1.5x
x[sup:32ih4xpe]2[/sup:32ih4xpe]-2.1x-2.7=0

This is where I lose it. I've used the quadratic formula to solve for x but I get x =2.31. What am I doing wrong?

Thank you.

#### stapel

##### Super Moderator
Staff member
crazylum said:
x[sup:22yj9xcz]2[/sup:22yj9xcz]-2.1x-2.7=0

This is where I lose it. I've used the quadratic formula to solve for x but I get x =2.31. What am I doing wrong?
Not being able to see what you did, I'm afraid it will be difficult for us to attempt to find where you might have made any errors. Sorry!

But the equation you have is correct, and the Quadratic Formula does indeed lead to the correct solution.

Eliz.

#### TchrWill

##### Full Member
crazylum said:
Henry and Irene working together can wash all the windows of their house in 1h 48m. Working alone, it takes Henry 1.5h more than Irene to do the job. How long does it take each person working alone to wash all the windows?
<< If it takes me 2 hours to paint a room and you 3 hours, ow long will it take to paint it together? >>

Method 1:

1--A can paint the house in 5 hours.
2--B can paint the house in 3 hours.
3--A's rate of painting is 1 house per A hours (5 hours) or 1/A (1/5) houses/hour.
4--B's rate of painting is 1 house per B hours (3 hours) or 1/B (1/3) houses/hour.
5--Their combined rate of painting is 1/A + 1/B (1/5 + 1/3) = (A+B)/AB (8/15) houses /hour.
6--Therefore, the time required for both of them to paint the 1 house is 1 house/(A+B)/AB houses/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.

Note - T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.

Method 2:

Consider the following diagram -

.........._______________ _________________
..........I B /............................/\
..........I..*.................../..............................I
..........I.....*............../................................I
..........Iy.......*........./.................................I
..........I................./...................................{
..........I*****x****** ....................................{
..........I............./....*................................(c)
..........I(c-y)..../.........*...............................{
..........I......../...............*...........................I.
..........I....../....................*........................I
..........I..../.........................*.....................I
..........I../.............................*...................{
.........I./___________________* ________\/__
A

1--Let c represent the area of the house to be painted.
2--Let A = the number of hours it takes A to paint the house.
3--Let B = the number of hours it takes B to paint the house.
4--A and B start painting at the same point but proceed in opposite directions around the house.
5--Eventually they meet in x hours, each having painted an area proportional to their individual painting rates.
6--A will have painted y square feet and B will have painted (c-y) square feet.
7--From the figure, A/c = x/y or Ay = cx.
8--Similarly, B/c = x/(c-y) or by = bc - cx.
9--From 7 & 8, y = cx/a = (bc - cx)/b from which x = AB/(A+B), one half of the harmonic mean of A and B.

I think this should give you enough of a clue as to how to solve your particular problem.

The key to the solution is that the combined time of two people is T = AB/(A + B) where A and B are the given rates of the individual people working alone.

In your case, Let I = Irene's time to complete the job alone and H = Henry's time to complete the job alone.
Given that Henry takes 1.5 hours more than Irene to complete the job, The combined time must therefore be
Tc = IH/(I + H) = I(I+1.5)/(I+I+1.5) = I^2 + 1.5I/(2I + 1.5) = 1.8 which simplifies to I^2 - 2.1I - 2.7 = 0.

I'll let you take it to its successful conclusion.

#### Subhotosh Khan

##### Super Moderator
Staff member
crazylum said:
My textbook gives the following word problem:

Henry and Irene working together can wash all the windows of their house in 1h 48m. Working alone, it takes Henry 1.5h more than Irene to do the job. How long does it take each person working alone to wash all the windows?

I know the answer is 3 and 4.5, but I don't know how they arrived at this. Here is what I do know and have done:

1h 48m = 1.8h

Irene's work time: x
Henry's work time: x + 1.5
Working together: 1/x + 1/(x+1.5) = 1/1.8

I've multiplied through by 1.8x(x+1.5) to eliminate the denominators and ended up with the following:

1.8(x+1.5) + 1.8x = x(x+1.5)
1.8x + 2.7 + 1.8x = x[sup:23h0o44q]2[/sup:23h0o44q]+1.5x
3.6x +2.7 = x[sup:23h0o44q]2[/sup:23h0o44q]+1.5x
x[sup:23h0o44q]2[/sup:23h0o44q]-2.1x-2.7=0 <<< This is Correct - you are making mistake somewhere trying to solve the quadratic equation

This is where I lose it. I've used the quadratic formula to solve for x but I get x =2.31. What am I doing wrong?

Thank you.

#### crazylum

##### New member
Figured it out. In the discriminant I was doing the following:
-2.1[sup:gi4cc31r]2[/sup:gi4cc31r] which was giving me -4.41 instead of (-2.1)[sup:gi4cc31r]2[/sup:gi4cc31r] which is 4.41. Silly mistake. Thanks for all of the replies.