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.