# Systems of Linear Equations

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

1. ### mathematicsstudentGuest

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 $12 per pound. How many pounds of each should be used? I figured out that the first equation should be: X + Y = 105 How do I write the other equation? I tried this, but it didn't work: 08x + .15Y = 12 If you can guide me along, I would greatly appreciate it! Thank you. mathematicsstudent, Jan 28, 2011 1. ### Advertisements 2. ### William ElliotGuest 08 looks like a mistake. Did you mean .08? No. More sense is 8x + 15y = 12. 8$/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 1. ### Advertisements 3. ### Barb KnoxGuest <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
4. ### The Qurqirish DragonGuest

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 5. ### Barb KnoxGuest 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 $24? My example is short, and even has a little bit of drama in it. -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum videtur. | BBB aa a r bbb | ----------------------------- Barb Knox, Jan 29, 2011 6. ### Stan BrownGuest Your example is just fine, but the original could have been fixed by simply changing "sells for" to "costs". Stan Brown, Jan 30, 2011 7. ### Alexandros BantisGuest There are a couple of questions that have to be answered. First, is how to express the two variables in terms of a single variable. Pounds(trailmix) + Pounds(cashews) = 105 Therefore P(cashews) = 105 - P(trailmix) Thus, we have our two variables: P = pounds of trailmix 105 - P = pounds of cashews Next we have to determine the weighted average. The basic formula would be Average price =$/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