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Math Help - Why do 9+4 work? Why does it give you 6?

  1. #1
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    Why do 9+4 work? Why does it give you 6?

    The question is:

    How can you bring up from a river, exactly 6 quarts of water, when you have only two containers, a four-quart pail and a nine-quart pail, to measure with. I understand how to get the answer to this but my question is why do 9 quart and 4 quart pails work? Why does it give you 6 quarts when you solve this question?
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  2. #2
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    Using the right combination  2\times 9 - 3\times 4 works. But what are you really asking?

    Do you want to know some combinations that don't work? Or you you looking for a generalisation?
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  3. #3
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    Thank you for your time. Yes I am looking for a generalization if you have any thoughts.
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  4. #4
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    this is like one of those "puzzle" problems. I assume all you have is one each 9qt and 4qt container.

    start with 9qt empty

    fill 4qt ... dump into 9qt

    fill 4qt again ... dump into 9 qt

    fill 4qt again ... top off 9qt. this leaves 3 qts in the 4 qt container.

    empty the 9qt ... pour the 3 qts into the 9qt from the 4qt container.

    fill 4qt ... pour into 9qt. now have 3+4 = 7 qts in the 9qt container.

    fill 4qt ... top off 9qt container. leaves 2 qts in the 4qt container.

    empty 9qt ... pour the 2qts into the 9qt container.

    fill the 4qt ... pour all 4 qts into the 9qt. 2 + 4 = 6 qts in the 9 qt container.
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  5. #5
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    Quote Originally Posted by loutja35 View Post
    The question is:

    How can you bring up from a river, exactly 6 quarts of water, when you have only two containers, a four-quart pail and a nine-quart pail, to measure with. I understand how to get the answer to this but my question is why do 9 quart and 4 quart pails work? Why does it give you 6 quarts when you solve this question?
    One generalisation going along w/ skeeter's reply is: If you have two pails, one that holds x quarts where x is a positive integer, and one that holds kx + 1 quarts for k a positive integer, you will be able to measure n quarts for any n in {0, ... , (k+1)x + 1}.
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  6. #6
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    This is wonderful. Thank you very much. This is what I was trying to understand.
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  7. #7
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    Here's a generalization: Suppose you have two pails, one that can hold exactly a quarts of water and another that can hold exactly b quarts of water.
    Furthermore, a and b are relatively prime and a<b . In this case, you can measure out any integer number of quarts 0\leq n\leq b of water, in the b -quarts pail.

    The way to do this parctically is described by the procedure skeeter gave.
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  8. #8
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by melese View Post
    Here's a generalization: Suppose you have two pails, one that can hold exactly a quarts of water and another that can hold exactly b quarts of water.
    Furthermore, a and b are relatively prime and a<b . In this case, you can measure out any integer number of quarts 0\leq n\leq b of water, in the b -quarts pail.

    The way to do this parctically is described by the procedure skeeter gave.
    I wanted to say this, but made a silly error mentally checking {9,5} and thought it somehow failed! Adding and subtracting single digit numbers can be challenging... especially late at night.
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  9. #9
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    Hello, loutja35!

    I too am puzzled by your question.
    But I'll give it a try . . .


    How can you measure 6 quarts of water, when you have only two containers:
    a 4-quart pail and a 9-quart pail to measure with?

    My question is: why do 9 quart and 4 quart pails work?
    Why does it give you 6 quarts when you solve this question?

    Answer: because a 4-quart pail and a 9-quart pail can be used
    . . . . . . to produce any integer quantity from 1 quart to 9 quarts.

    Let A = 9-quart pail.
    Let B = 4-quart pail.


    Fill \,A.
    Code:
          *   *
          |:::|
          |:::|
          |:::|   *   *
          |:9:|   |   |     We have 9 quarts.
          |:::|   |   |
          |:::|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Pour \,A into \,B.
    Code:
          *   *
          |   |
          |   |
          |:::|   *   *
          |:::|   |:::|     We have 5 quarts.
          |:5:|   |:::| 
          |:::|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Empty \,A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |:::|
          |   |   |:::|     We have 4 quarts.
          |   |   |:4:|
          |   |   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |:::|   |   |
          |:::|   |   |
          |:4:|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |:::|   |:::|
          |:::|   |:::|
          |:4:|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |:::|
          |:::|
          |:::|   *   *
          |:8:|   |   |     We have 8 quarts.
          |:::|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |:::|
          |:::|   *   *
          |:8:|   |:::|
          |:::|   |:::|
          |:::|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |:::|
          |:::|
          |:::|   *   *
          |:9:|   |   |
          |:::|   |:::|    We have 3 quarts.
          |:::|   |:3:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Empty \,A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |   |
          |   |   |:::|
          |   |   |:3:|
          |   |   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |   |
          |:::|   |   |
          |:3:|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |:::|
          |:::|   |:::|
          |:3:|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |   |
          |:::|   *   *
          |:::|   |   |
          |:7:|   |   |     We have 7 quarts.
          |:::|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |   |
          |:::|   *   *
          |:::|   |:::|
          |:7:|   |:::|
          |:::|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |:::|
          |:::|
          |:::|   *   *
          |:9:|   |   |
          |:::|   |   |     We have 2 quarts.
          |:::|   |:::|
          |:::|   |:2:|
          *---*   *---*
            A       B

    Empty \,A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |   |
          |   |   |   |
          |   |   |:::|
          |   |   |:2:|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |   |
          |   |   |   |
          |:::|   |   |
          |:2:|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |   |
          |   |   *   *
          |   |   |:::|
          |   |   |:::|
          |:::|   |:4:|
          |:2:|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |   |
          |   |
          |:::|   *   *
          |:::|   |   |
          |:6:|   |   |     We have 6 quarts.
          |:::|   |   |
          |:::|   |   |
          *---*   *---*
            A       B

    Fill \,B.
    Code:
          *   *
          |   |
          |   |
          |:::|   *   *
          |:::|   |:::|
          |:6:|   |:::|
          |:::|   |:4:|
          |:::|   |:::|
          *---*   *---*
            A       B

    Pour \,B into A.
    Code:
          *   *
          |:::|
          |:::|
          |:::|   *   *
          |:9:|   |   |     We have 1 quart.
          |:::|   |   |
          |:::|   |   |
          |:::|   |:1:|
          *---*   *---*
            A       B

    We have devised a procedure to get 1, 2, 3, 4, 5, 6, 7, and 8 quarts.
    . . The problem is completely solved.

    Why are you asking "Why?"

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  10. #10
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    Could you actually further explain your reasoning and how you decided on kx+1. What exactly does this stand for? Also how did you come up with (o,....,(k+1)x+1). How does this specifically work? Could you show me an exampe of this. As always, thank you very much!
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  11. #11
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    Quote Originally Posted by loutja35 View Post
    Could you actually further explain your reasoning and how you decided on kx+1. What exactly does this stand for? Also how did you come up with (o,....,(k+1)x+1). How does this specifically work? Could you show me an exampe of this. As always, thank you very much!
    For {9,4} note that at the beginning you fill the 4 quart pail to the top, so there are 4 quarts in it.

    After topping the 9 quart pail off, you are left with 3 quarts in the 4 quart pail.

    After topping the 9 quart pail off again, you are left with 2 quarts in the 4 quart pail.

    After topping the 9 quart pail off again, you are left with 1 quart in the 4 quart pail.

    This allows you to get any number of quarts in the 9 quart pail up to 9 quarts. (Think about why.)

    Note that if the large pail has capacity of any number in the set {5,9,13,17,21,...} the effect is the same; we can get any number of quarts up to the capacity of the pail. (Think about why.)

    Furthermore the same thing happens whenever the small pail is x and the larger pail is a (positive) multiple of x plus 1. (Think about why.)

    Furthermore there is no need for the number of quarts left in the small pail to be the sequence x, x-1, ... , 1. As long as every number in {1, 2, ... , x} can be obtained in the small pail, any number up to capacity of large pail can be gotten in large pail. This is true if and only if gcd(x, y) = 1 where y is the capacity of larger pail.
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  12. #12
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    A more advanced/thorough treatment can be found in this thread

    http://www.mathhelpforum.com/math-he...em-155981.html
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