document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30976>Question 30976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Joe has $40,000 to invest.  He invests part of it at 5%, one-fourth of this amount at 6%, and the rest of the money at 7%.  His total annual interest income is $2,530.  Find the amount 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 #17701 by </B><A HREF=/tutors/aboutme.mpl?userid=blubunny01><B>blubunny01(20)</B></A><img src=\"/images/stars/stars-06.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=blubunny01><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=17701\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's first define three variables to keep track of the amounts invested at each rate.\r<BR>\n" );
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document.write( "x = amount invested at 5%<BR>\n" );
document.write( "y = amount invested at 6%<BR>\n" );
document.write( "z = amount invested at 7%\r<BR>\n" );
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document.write( "This problem has three unknown values - let's establish the relationships between these unknowns.\r<BR>\n" );
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document.write( "We know that the amount invested at 6% (y) is 1/4 the amount invested at 5% (x).  Therefore, we can say:<BR>\n" );
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document.write( "Then, whatever is not invested at 5% or 6% will be invested at 7%.  The total amount of money Joe is investing is $40,000.  So, we can say:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+40000+-+x+-+y ALIGN=MIDDLE  ALT=\"z+=+40000+-+x+-+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to find the \"rest of the money\" invested at 7%)<BR>\n" );
document.write( " or...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+40000+-+x+-+%281%2F4%29%2Ax ALIGN=MIDDLE  ALT=\"z+=+40000+-+x+-+%281%2F4%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (substituting the y equation we created earlier)\r<BR>\n" );
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document.write( "We know his total annual interest income is $2,530.  To find the income, we just take the amounts invested at each rate and multiply them by their respective rates (i.e. since x is the amount invested at 5%, we find the interest income by multiplying 0.05x).  This means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05%2Ax+%2B+0.06%2Ay+%2B+0.07%2Az+=+2530 ALIGN=MIDDLE  ALT=\"0.05%2Ax+%2B+0.06%2Ay+%2B+0.07%2Az+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " or...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05%2Ax+%2B+0.06%2A%281%2F4%29%2Ax+%2B+0.07%2A%2840000+-+x+-+%281%2F4%29%2Ax%29+=+2530 ALIGN=MIDDLE  ALT=\"0.05%2Ax+%2B+0.06%2A%281%2F4%29%2Ax+%2B+0.07%2A%2840000+-+x+-+%281%2F4%29%2Ax%29+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, solve for x:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05%2Ax+%2B+0.015%2Ax+%2B+0.07%2840000+-+1.25%2Ax%29+=+2530 ALIGN=MIDDLE  ALT=\"0.05%2Ax+%2B+0.015%2Ax+%2B+0.07%2840000+-+1.25%2Ax%29+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [simplify equation]\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05%2Ax+%2B+0.015%2Ax+%2B+0.07%2840000+-+1.25%2Ax%29+=+2530 ALIGN=MIDDLE  ALT=\"0.05%2Ax+%2B+0.015%2Ax+%2B+0.07%2840000+-+1.25%2Ax%29+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [simplify equation]\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.065%2Ax+%2B+2800+-+0.0875%2Ax+=+2530 ALIGN=MIDDLE  ALT=\"0.065%2Ax+%2B+2800+-+0.0875%2Ax+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [more simplifying the equation]\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2800+-+0.0225%2Ax+=+2530 ALIGN=MIDDLE  ALT=\"2800+-+0.0225%2Ax+=+2530\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [combine like terms]\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.0225%2Ax+=+-+270 ALIGN=MIDDLE  ALT=\"-0.0225%2Ax+=+-+270\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [combine like terms]\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+12%2C000 ALIGN=MIDDLE  ALT=\"x+=+12%2C000\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [divide both sides by -0.0225 to isolate x]\r<BR>\n" );
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document.write( "Going back to our original equations:\r<BR>\n" );
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document.write( "x = amount invested at 5%<BR>\n" );
document.write( "y = amount invested at 6% = (1/4)x<BR>\n" );
document.write( "z = amount invested at 7% = 40,000 - x - y\r<BR>\n" );
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document.write( "We have x = 12,000.  <BR>\n" );
document.write( "y = (1/4)(12,000) = 3,000.<BR>\n" );
document.write( "z = 40,000 - 12,000 - 3,000 = 25,000. </SPAN>\n" );
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