document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=91561>Question 91561</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A TOTAL OF 27,000 IS INVESTED , PART OF IT AT 12% PART OF IT AT 13%. THE TOTAL INTEREST AFTER A YEAR IS 3,385 . HOW MUCH WAS INVESTED AT EACH RATE? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #66501 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=66501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are two unknowns in this problem ... the last sentence (How much was invested at<BR>\n" );
document.write( "each rate?) tells you this.<BR>\n" );
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document.write( "So let's begin by identifying the two unknowns ... call the amount invested at 12% A, and call<BR>\n" );
document.write( "the amount invested at 13% B.<BR>\n" );
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document.write( "Now we can begin writing equations.  For the first equation we know that the total amount<BR>\n" );
document.write( "invested is $27,000. But the total amount invested is A plus B. Setting these two equal we<BR>\n" );
document.write( "get a first equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+B+=+27000 ALIGN=MIDDLE  ALT=\"A+%2B+B+=+27000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since A is the amount invested at 12% (or its decimal equivalent 0.12) we know that the<BR>\n" );
document.write( "amount of interest made on this investment is 0.12 times A. And since B is the amount <BR>\n" );
document.write( "invested at 13% (or at 0.13) we know that the amount of interest made on this investment<BR>\n" );
document.write( "is 0.13 times B. The sum of these two amounts is 0.12A + 0.13B, and the problem tells you<BR>\n" );
document.write( "that the total amount of interest for the year is $3385.  Therefore we have another<BR>\n" );
document.write( "equation based on the total interest. This equation says:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.12A+%2B+0.13B+=+3385 ALIGN=MIDDLE  ALT=\"0.12A+%2B+0.13B+=+3385\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since you can't solve a single equation with two unknowns, we need to find a way of combining<BR>\n" );
document.write( "our two equations so that we end up with a single equation having only one unknown.<BR>\n" );
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document.write( "One way we can do that is to solve our first equation for one of the unknowns in terms<BR>\n" );
document.write( "of the other unknown, and substitute that result into our second equation.  Our first<BR>\n" );
document.write( "equation said:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+B+=+27000 ALIGN=MIDDLE  ALT=\"A+%2B+B+=+27000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract B from both sides of this equation and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+27000+-+B ALIGN=MIDDLE  ALT=\"A+=+27000+-+B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now let's go to our second equation which said:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.12A+%2B+0.13B+=+3385 ALIGN=MIDDLE  ALT=\"0.12A+%2B+0.13B+=+3385\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But we have learned that A equals 27000 - B. So in our second equation we can replace<BR>\n" );
document.write( "A with 27000 - B to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.12%2827000+-+B%29+%2B+0.13B+=+3385 ALIGN=MIDDLE  ALT=\"0.12%2827000+-+B%29+%2B+0.13B+=+3385\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that this \"new\" equation has only one unknown ... B. Therefore, we can solve it<BR>\n" );
document.write( "for B.  Begin by multiplying 0.12 times each of the terms in the parentheses.  When you do<BR>\n" );
document.write( "that the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3240+-+0.12B+%2B+0.13B+=+3385 ALIGN=MIDDLE  ALT=\"3240+-+0.12B+%2B+0.13B+=+3385\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combining the two terms that contain a \"B\" reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3240+%2B+0.01B+=+3385 ALIGN=MIDDLE  ALT=\"3240+%2B+0.01B+=+3385\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then get rid of the 3240 on the left side by subtracting 3240 from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01B+=+3385+-+3240+=+145 ALIGN=MIDDLE  ALT=\"0.01B+=+3385+-+3240+=+145\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So our equation is now:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01B+=+145 ALIGN=MIDDLE  ALT=\"0.01B+=+145\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can solve for B by dividing both sides of this equation by 0.01 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=+145%2F0.01+=+14500 ALIGN=MIDDLE  ALT=\"B+=+145%2F0.01+=+14500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we know that B (the amount invested at 13%) is $14,500. And since the total amount<BR>\n" );
document.write( "invested was $27,000, the remaining $12,500 must be the amount invested at 12%.<BR>\n" );
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document.write( "Let's check:<BR>\n" );
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document.write( "$14,500 invested at 13% would earn 14500*0.13 = $1885<BR>\n" );
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document.write( "And $12,500 invested at 12% would earn 12500*0.12 = $1500<BR>\n" );
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document.write( "So the total earnings would be $1885 + $1500 = $3385<BR>\n" );
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document.write( "This is exactly the amount that the problem said it should be, so that checks. <BR>\n" );
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document.write( "And the total amount invested is $14,500 + $12,500 = $27,000. Again, this is exactly<BR>\n" );
document.write( "the amount the problem said it was. So our answers check all the way.<BR>\n" );
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document.write( "The answers are that $14,500 is the amount invested at 13% and $12,500 at 12%.<BR>\n" );
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document.write( "Hope this helps you to understand the problem.<BR>\n" );
document.write( " </SPAN>\n" );
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