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A tricky one!

  1. Aug 4, 2003 #1
    Hey guys.

    I have this tricky math question.

    Peter and his big sister Lisa are together 11 years old. The product of their age is equal to the age of their father. When Peter was born the father was 8 times older than Lisa. How old are Lisa and Peter?

    You can put up the info in the text like this:
    1. Peter + Lisa = 11
    2. Peter * Lisa = Fathers age
    3. Fathers age = Lisa * 8

    Well I have tried to solve this but without any help.

    Anybody know how to do this?

    Thanks.
     
  2. jcsd
  3. Aug 4, 2003 #2

    HallsofIvy

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    You've made a good start but the last equation is wrong.


    The last sentence does not say that the father's age NOW is 8 times is 8 times Lisa's age- and that is what you are using "father's age", "Peter" and "Lisa" to represent in the first two equations.

    If "Peter" is used to represent Peter's age now, then Peter was born "Peter" years ago (Yes, that looks silly- that's one reason why it's often better to use letters rather than words!).

    The father's age when Peter was born was:
    "father's age- Peter". At that time Lisa's age was "Lisa- Peter".
    The last equation is "father- Peter= 8*(Lisa- Peter)"

    Here's how I would do the problem myself:

    Let P= Peter's age now, L= Lisa's age now, F= father's age now.

    Then the first sentence, "Peter and his big sister Lisa are together 11 years old.", becomes the equation "P+ L= 11".

    The second sentence, "The product of their age is equal to the age of their father.", becomes "L*P= F".

    These two equations are exactly the same as your first two equations.

    The third sentence, "When Peter was born the father was 8 times older than Lisa.", becomes "F- P= 8*(L- P)".
    That last equation is equivalent to F- P= 8L- 8P which is also equivalent to F= 8L- 7P.

    The first equation, P+ L= 11 is the same as L= 11- P (Subtract P from both sides).

    If we replace the L in F= 8L- 7P by 11-P, we get F= 8(11-P)-7P
    = 88- 8P- 7P= 88- 15P: that is, F= 88- 15P.

    Now, replace F and L in L*P= F by those:

    L*P= (11-P)*P= 88- 15P which is the same as
    11P- P<sup>2</sup>= 88- 15P

    adding P<sup>2</sup> and subtracting 11P from both sides gives

    P<sup>2</sup>- 26P+ 88= 0.

    That's a quadratic equation for P. You can solve that by using the quadratic equation, completing the square, or factoring (factoring is simplest IF the polynomial factors- here it DOES!).

    Once you have found Peter's age (P), you can use L= 11- P to find Lisa's age and F= 88- 15P to find the father's age.

    Be sure to check that your answers make the original sentences true!
     
  4. Aug 4, 2003 #3
    Ok, I didn't understand the previous solution, so I'll post my own... but I've always been horrible at maths...

    Peter's age - x.
    Lisa's age - y.
    Father's age - z.

    x + y = 11
    xy = z
    8 (y - x) + x = z

    xy = 8y - 7x
    11x - x^2 (squared) = 88 - 8x - 7x
    x^2 - 26x + 88 = 0
    x1 = 22 (doesn't fit)
    x2 = 4

    y = 11 - 4 = 7
    4 * 7 = z = 28

    Peter's age = x = 4
    Lisa's age = y = 7
    Father's age = z = 28
     
    Last edited: Aug 5, 2003
  5. Aug 4, 2003 #4
    BTW

    (y - x) is the difference in age between lisa and peter
     
  6. Aug 5, 2003 #5
    thanks allot guys.
    I really could use you help, and I appreciate it!
     
  7. Aug 5, 2003 #6
    Tail
    Umm ... weird.
    Your third equation is wrong, it should be :
    8(y-x)=(z-x)
    BUT[/B} the first line in your solution (which is supposed to be based on the second and third equations) is right, so i suppose you just made a mistake while writing the third equation.
    Otherwise, your answers are right :smile:.
     
  8. Aug 5, 2003 #7
    I edited the post... a little mistake, as I did the calculations on paper and then typed them...
     
  9. Aug 5, 2003 #8

    HallsofIvy

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    Staii--- Have you been staying up too late?

    Tail's "third equation" 8(y- x)+ x= z is exactly equivalent to
    8(y-x)= z- x (and to my "F- P= 8*(L- P)").

    True, a direct translation of the sentence gives 8(y-x)= z- x but
    Tail just went ahead and added x to both sides.
     
  10. Nov 18, 2009 #9
    PETER is 4
    LISA is 7
    Their father was 24 when Peter was born, but at that time Lisa was 3.
    Then 4 years later Lisa is 7 and Peter is 4.
    Now since it is 4 years later their father is 28
    7 times 4 is equal to 28.
     
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