document.write( "Question 102426: I looked over my notes several times, but I still have trouble setting up the problem and solving it. I'm not even sure I have this under the right category. This is being solved using a Quantity-Value Table.\r
\n" ); document.write( "\n" ); document.write( "Juan has coffee that sells for $9 a pound and $4 a pound. How many pounds of each must be mixed to get 20 pounds of coffee worth $8.25 a pound?\r
\n" ); document.write( "\n" ); document.write( "I know the table is set up as Q|V|Q*V. What I tried using was x*4, x*9, and 20*8.25. I'm not sure I have it set up correctly. Then I got to 9x+4x=165. Solve and get x = 165/13. I know I'm supposed to apply that answer to get the resulting answer for both the $9 a pound and $4 a pound, but I'm not sure how.\r
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Algebra.Com's Answer #74421 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
Let's pick this problem apart. We'll start by recognizing that we need to find the weight of
\n" ); document.write( "two different coffees. There is $4 a pound coffee, so let's call its unknown weight F
\n" ); document.write( "(standing for 4). There is also $9 a pound coffee, so let's call its unknown weight N
\n" ); document.write( "(for 9).
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\n" ); document.write( "Common sense will tell us that the most we can sell coffee for is $9 per pound. Since the
\n" ); document.write( "coffee we want to mix up is going to cost $8.25 per pound, we're going to need more of the
\n" ); document.write( "$9 per pound stuff in the mix than the $4 a pound cheap stuff. In fact if we mixed equal weights
\n" ); document.write( "of the $9 coffee and $4 coffee, we should get coffee that sells for $6.50 a pound -- halfway
\n" ); document.write( "between the two prices. I'm throwing this info in just as a mental check we can use later
\n" ); document.write( "to make sure our answer makes sense.
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\n" ); document.write( "We want to get 20 pounds of mixed coffee. Therefore, we know that the sum of the two weights
\n" ); document.write( "F and N must equal 20. So we can write our first equation as:
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\n" ); document.write( "F + N = 20
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\n" ); document.write( "That's pretty straightforward. No magic there.
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\n" ); document.write( "Now let's start looking at the cost. If we have F pounds of coffee at $4 per pound, then
\n" ); document.write( "the value of the $4 coffee we have in the mix is $4 times the number of pounds F.
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\n" ); document.write( "Similarly, if we have N pounds of coffee in the mix at $9 per pound, then the total value of
\n" ); document.write( "the $9 per pound coffee in the mix is $9 times the number of pounds which is N.
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\n" ); document.write( "Now how much is the mix worth. We know there are 20 pounds of mix and it sells for $8.25
\n" ); document.write( "per pound. So we multiply 20 times $8.25 and get $165 dollars.
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\n" ); document.write( "Now let's add up the values of the two ingredients and we can set that equal to the value
\n" ); document.write( "of the mix. The values of the two ingredients is $4*F and $9*N and these combined amounts
\n" ); document.write( "must equal the $165 that the mix is worth. In equation form this is:
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\n" ); document.write( "4F + 9N = 165
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\n" ); document.write( "So we have two equations to work with:
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\n" ); document.write( "F + N = 20 and
\n" ); document.write( "4F + 9N = 165
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\n" ); document.write( "We can solve this set a number of ways. Let's use variable elimination. Suppose we multiply
\n" ); document.write( "the top equation [both sides and all terms] by -4. If we do that the equation set becomes:
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\n" ); document.write( "-4F - 4N = -80 and
\n" ); document.write( "+4F + 9N = 165
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\n" ); document.write( "Now vertically in columns let's add the two equations together. Notice that the -4F in the
\n" ); document.write( "top equation cancels the +4F in the bottom equation. [We deliberately made that happen by
\n" ); document.write( "multiplying the top equation by -4.]
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\n" ); document.write( "The -4N and the +9N add together to give +5N and the -80 and +165 add together to give +85.
\n" ); document.write( "So after adding these two equations we are left with the single equation:
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\n" ); document.write( "5N = 85
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\n" ); document.write( "Solve for N by dividing both sides by 5 to get:
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\n" ); document.write( "N = 85/5 = 17
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\n" ); document.write( "So we need 17 pounds of the $9 a pound coffee. Since we need a total of 20 pounds, this
\n" ); document.write( "means that we must have 3 pounds of the $4 per pound coffee.
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\n" ); document.write( "Just as we \"guessed\" earlier, we needed a lot more of the $9 coffee in the mix than we did
\n" ); document.write( "of the $4 coffee.
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\n" ); document.write( "Hope this helps you to see the general way that you can approach problems such as these.
\n" ); document.write( "You need two equations to solve for two unknowns, so you can start on that basis.
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