algebra

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Here's the problem: It takes Stan 20 days to build a house by himself. When his wife, Brenda, joins him, it takes only 14 days. How long does it take Brenda to build a house on her own?

Is there some sort of formula to solve this? I have no idea where to begin. Thanks for the help!

  • algebra -

    Consider the following:

    << 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.

  • algebra -

    Thank you!

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