document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110049>Question 110049</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coffee merchant has coffee beans that sell for $9 per pound and $12 per pound. The two types are to be mixed to create 100 lb of a mixture that will sell for $11.25 per pound. How much of each type of bean should be used in the mixture?\r<BR>\n" );
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document.write( "Thank you for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80190 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=80190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's pick this problem apart one step at a time. We'll begin by letting N represent the <BR>\n" );
document.write( "the number of pounds of coffee at Nine dollars a pound and T represent the number of pounds<BR>\n" );
document.write( "of coffee at Twelve dollars a pound.<BR>\n" );
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document.write( "As a rough check we can tell that T must be more than N. Why is that? because the final mixture<BR>\n" );
document.write( "must sell at $11.25 a pound ... and this price is closer to $12 a pound than it is to $9.00 a pound.<BR>\n" );
document.write( "[If N equaled T, we would expect the coffee mix to sell for $10.50 a pound, which is the price<BR>\n" );
document.write( "that is halfway between $9.00 a pound and $12.00 a pound.]<BR>\n" );
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document.write( "Now to the problem.  The total weight of the mix must be 100 pounds. Therefore, we know that<BR>\n" );
document.write( "N + T must equal 100.  In equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+%2B+T+=+100 ALIGN=MIDDLE  ALT=\"N+%2B+T+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next, we can tell that at $11.25 per pound the entire 100 pounds of mix must be worth<BR>\n" );
document.write( "$11.25 times 100 which equals $1125.00.<BR>\n" );
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document.write( "Of that total amount, the value of the $9.00 per pound coffee is $9 times the number of<BR>\n" );
document.write( "pounds of that coffee (which is N). So 9*N is the dollar amount of the $9 coffee in the mix.<BR>\n" );
document.write( "And the value of the of the $12.00 per pound coffee in the mix is 12*T. The combined<BR>\n" );
document.write( "value of the two coffees in the mix must add to $1125.00 and in equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9N+%2B+12T+=+1125 ALIGN=MIDDLE  ALT=\"9N+%2B+12T+=+1125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By itself, this equation cannot be solved because it is a single equation and there are<BR>\n" );
document.write( "two unknowns ... N and T. But we can go back to the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+%2B+T+=+100 ALIGN=MIDDLE  ALT=\"N+%2B+T+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and solve for one of the variables in terms of the other. For example, let's solve for<BR>\n" );
document.write( "N by subtracting T from both sides of the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+100+-+T ALIGN=MIDDLE  ALT=\"N+=+100+-+T\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The right side of this equation is equal to N so it can be substituted for N in the dollar<BR>\n" );
document.write( "equation. Start with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9N+%2B+12T+=+1125 ALIGN=MIDDLE  ALT=\"9N+%2B+12T+=+1125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and replace N with 100 - T to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2A%28100+-+T%29+%2B+12T+=+1125 ALIGN=MIDDLE  ALT=\"9%2A%28100+-+T%29+%2B+12T+=+1125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the left side to make it:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900+-+9T+%2B+12T+=+1125 ALIGN=MIDDLE  ALT=\"900+-+9T+%2B+12T+=+1125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the 900 on the left side by subtracting 900 from both sides to reduce the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9T+%2B+12T+=+225 ALIGN=MIDDLE  ALT=\"-9T+%2B+12T+=+225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combine the two terms on the left side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3T+=+225 ALIGN=MIDDLE  ALT=\"3T+=+225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for T by dividing both sides by 3 and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+225%2F3+=+75 ALIGN=MIDDLE  ALT=\"T+=+225%2F3+=+75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells you that the 100 pounds of mix contains 75 pounds of the coffee that costs $12 per pound.<BR>\n" );
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document.write( "The remainder of the 100 pounds of mix must therefore be 25 pounds of the coffee that costs<BR>\n" );
document.write( "$9 per pound.<BR>\n" );
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document.write( "Check ... 25 pounds + 75 pounds = 100 pounds of mix.<BR>\n" );
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document.write( "And at $9 per pound, the 25 pounds of it in the mix is worth $9 * 25 = $225. Plus at<BR>\n" );
document.write( "$12 per pound, the 75 pounds of it in the mix is worth $12 * 75 = $900. So the total worth<BR>\n" );
document.write( "of the mix is $225 + $900 = $1125. Finally there are 100 lbs of the mix and at a total worth<BR>\n" );
document.write( "of $1125, the cost per pound should be $1125/100 = $11.25 and this is what the problem said<BR>\n" );
document.write( "it should be.<BR>\n" );
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document.write( "Everything checks. Therefore, you can say that the required mix consists of 25 pounds of<BR>\n" );
document.write( "the $9 per pound coffee and 75 pounds of the $12 per pound coffee. <BR>\n" );
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document.write( "Finally, note that there is more of the $12 per pound coffee in the mix than there is<BR>\n" );
document.write( "$9 per pound coffee just as we originally thought.<BR>\n" );
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document.write( "Hope this helps you to understand the problem.<BR>\n" );
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