Systems of Linear Equations

Discussion in 'Undergraduate Math' started by mathematicsstudent, Jan 28, 2011.

  1. Hello,

    I have to solve this problem:

    The manager of a bulk foods establishment sells a trail mix for $8 per pound and premium cashews for $15 per pound. The manager wishes to make a 105-pound trail mix-cashew mixture that will sell for $12 per pound. How many pounds of each should be used?

    I figured out that the first equation should be:
    X + Y = 105

    How do I write the other equation? I tried this, but it didn't work:
    08x + .15Y = 12

    If you can guide me along, I would greatly appreciate it!

    Thank you.
     
    mathematicsstudent, Jan 28, 2011
    #1
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  2. 08 looks like a mistake. Did you mean .08?
    No. More sense is 8x + 15y = 12.

    8$/lb * x lb + 15$/lb * y lb = 12$/lb

    The dimensions aren't correct.
    The left hand side is dollars and the
    right hand side is dollars per pound.

    Where did the notion of 15%, ie .15 come from?
    8$/lb * x lb + 15$/lb * y lb = 12$/lb * how many lbs?
    = cost of the mix.
     
    William Elliot, Jan 29, 2011
    #2
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  3. mathematicsstudent

    Barb Knox Guest

    <rant>

    Yet another wretched attempt to make a mathematics problem seem relevant
    to the "real world". In the actual real world, the price a retailer
    charges for a mixture is NOT determined by the prices charged for the
    separate ingredients. In the actual real world, the retailer would be
    more likely to put in about 20% cashews and charge $11.95 per pound for
    the New Cashew-Enhanced Trail Mix.

    The really sad aspect of this is that with a half a minute of thought
    the problem writer could have come up with a mixture problem that did
    make sense in the "real world".

    </rant>

    [snip]
    --
    ---------------------------
    | BBB b \ Barbara at LivingHistory stop co stop uk
    | B B aa rrr b |
    | BBB a a r bbb | Quidquid latine dictum sit,
    | B B a a r b b | altum videtur.
    | BBB aa a r bbb |
    -----------------------------
     
    Barb Knox, Jan 29, 2011
    #3
  4. And similarly the cost of packaging is different, and then you have
    increased cost from buying smaller quantities of the two types before
    mixing, and the shopkeep is greedy, and there is another store across
    the street selling an unknown mixture of these for $11, and...
    I hope you get the point of that. The use of nuts and trail mix is NOT
    to give a real-world example, but rather to make something concrete
    that people who aren't pure math thinkers can hopefully grasp more
    easily.
    Word this problem anyway you want, and I can find something that
    doesn't match the "real world"
    Even if you want to remain reasonable, they don't want to use 3 pages
    in a book for one word problem.
     
    The Qurqirish Dragon, Jan 29, 2011
    #4
  5. mathematicsstudent

    Barb Knox Guest

    I accept your challenge:

    Barbara has the job of providing some nibblies for a party, with a
    strict budget of $24 for 2 pounds mixed nuts which must include cashews,
    since that's the host's favourite. The local bulk food store does have
    mixed nuts at $8 per pound (tax inclusive), but these lack cashews.
    They also have just cashews at $15 per pound (tax inclusive). What
    quantity of each of those two items should she buy in order to make 2
    pounds altogether for $24?

    My example is short, and even has a little bit of drama in it.

    --
    ---------------------------
    | BBB b \ Barbara at LivingHistory stop co stop uk
    | B B aa rrr b |
    | BBB a a r bbb | Quidquid latine dictum sit,
    | B B a a r b b | altum videtur.
    | BBB aa a r bbb |
    -----------------------------
     
    Barb Knox, Jan 29, 2011
    #5
  6. mathematicsstudent

    Stan Brown Guest

    Your example is just fine, but the original could have been fixed by
    simply changing "sells for" to "costs".
     
    Stan Brown, Jan 30, 2011
    #6
  7. There are a couple of questions that have to be answered. First, is how
    to express the two variables in terms of a single variable.

    Pounds(trailmix) + Pounds(cashews) = 105
    Therefore P(cashews) = 105 - P(trailmix)
    Thus, we have our two variables:

    P = pounds of trailmix
    105 - P = pounds of cashews

    Next we have to determine the weighted average. The basic formula would
    be Average price = $/pounds.

    Using dimensional analysis, we can arrive at the weighted average as
    follows:

    Average Price = ($/lb) / Total Pounds

    This is because the 'pounds' cancel each other out, and you are left
    with price.

    hope this helps,

    alex
     
    Alexandros Bantis, Jan 30, 2011
    #7
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