document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187174>Question 187174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a business invests $8,000 in a savings account for 2 years. at the beginning of teh second year, an additional $2500 is invested. At the end of the second year, the account balance is $ 11445. What was the annual interest rate?\r<BR>\n" );
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document.write( "So far I used the compounding interest formula where I = P(1 + r)^t<BR>\n" );
document.write( "for the 1 st year's interest = (8000 + 8000r)<BR>\n" );
document.write( "and for the second year's interest was (1st year's interest + [(8000+2500) + (8000r)(1 + r)^1] = total income after two years of 11445.\r<BR>\n" );
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document.write( "to set up my quadratic equation I combined like terms<BR>\n" );
document.write( "8000 + 8000r + [(10500 + 8000r)(1 + r)] = 11445<BR>\n" );
document.write( "8000 + 8000r + [10500 + 8000r + 10500r + 8000r^2] = 11445<BR>\n" );
document.write( "combining terms 18500 + 26500r + 8000r^2 = 11445<BR>\n" );
document.write( "or 8000r^2 + 26500r  = 11445 - 18500 or -7055<BR>\n" );
document.write( "8000r^2 + 26500r + 7055 = 0\r<BR>\n" );
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document.write( "at this point I get stuck because the numbers are too high in order to effectively use the quadratic formula to solve for r\r<BR>\n" );
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document.write( "I believe the answer is .05 or 5% annual interest rate..can you show me how you worked through the quadratic formula...<BR>\n" );
document.write( "Thank you very much! \r<BR>\n" );
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document.write( "Kerry </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140304 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=140304\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%29 ALIGN=MIDDLE  ALT=\"A=P%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the interest formula. \r<BR>\n" );
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document.write( "Note: since we're only looking at intervals of a year, this means that the value of \"t\" is 1 and simplifies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%29%5Et ALIGN=MIDDLE  ALT=\"A=P%281%2Br%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%29%5E1 ALIGN=MIDDLE  ALT=\"A=P%281%2Br%29%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%29 ALIGN=MIDDLE  ALT=\"A=P%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=8000%281%2Br%29 ALIGN=MIDDLE  ALT=\"A=8000%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=8000 ALIGN=MIDDLE  ALT=\"P=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the initial investment) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we want to find the amount at the end of the first year)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=8000%2B8000r ALIGN=MIDDLE  ALT=\"A=8000%2B8000r\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "So after one full year, you now have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8000%2B8000r ALIGN=MIDDLE  ALT=\"8000%2B8000r\"   DISABLED_style= \"max-width:90%; height:auto;\" > dollars in the account.\r<BR>\n" );
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document.write( "Now because \"an additional $2,500 is invested\", this pushes up the amount to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8000%2B8000r%2B2500 ALIGN=MIDDLE  ALT=\"8000%2B8000r%2B2500\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just add 2500 to the last expression). \r<BR>\n" );
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document.write( "Combine like terms to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10500%2B8000r ALIGN=MIDDLE  ALT=\"10500%2B8000r\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "So the new principal is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=10500%2B8000r ALIGN=MIDDLE  ALT=\"P=10500%2B8000r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this principal will be invested in the account for year #2)\r<BR>\n" );
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document.write( "Since after two years you have $11,445, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=11445 ALIGN=MIDDLE  ALT=\"A=11445\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%29 ALIGN=MIDDLE  ALT=\"A=P%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the original formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11445=%2810500%2B8000r%29%281%2Br%29 ALIGN=MIDDLE  ALT=\"11445=%2810500%2B8000r%29%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=11445 ALIGN=MIDDLE  ALT=\"A=11445\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=10500%2B8000r ALIGN=MIDDLE  ALT=\"P=10500%2B8000r\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11445=10500%2B18500%2Ar%2B8000%2Ar%5E2 ALIGN=MIDDLE  ALT=\"11445=10500%2B18500%2Ar%2B8000%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=10500%2B18500%2Ar%2B8000%2Ar%5E2-11445 ALIGN=MIDDLE  ALT=\"0=10500%2B18500%2Ar%2B8000%2Ar%5E2-11445\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 11445 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=8000%2Ar%5E2%2B18500%2Ar-945 ALIGN=MIDDLE  ALT=\"0=8000%2Ar%5E2%2B18500%2Ar-945\"   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=8000%2Ar%5E2%2B18500%2Ar-945=0 ALIGN=MIDDLE  ALT=\"8000%2Ar%5E2%2B18500%2Ar-945=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2%2Bbr%2Bc ALIGN=MIDDLE  ALT=\"ar%5E2%2Bbr%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=8000 ALIGN=MIDDLE  ALT=\"a=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=18500 ALIGN=MIDDLE  ALT=\"b=18500\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-945 ALIGN=MIDDLE  ALT=\"c=-945\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for r\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"r+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-%2818500%29+%2B-+sqrt%28+%2818500%29%5E2-4%288000%29%28-945%29+%29%29%2F%282%288000%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=8000 ALIGN=MIDDLE  ALT=\"a=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=18500 ALIGN=MIDDLE  ALT=\"b=18500\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-945 ALIGN=MIDDLE  ALT=\"c=-945\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+sqrt%28+342250000-4%288000%29%28-945%29+%29%29%2F%282%288000%29%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+sqrt%28+342250000-4%288000%29%28-945%29+%29%29%2F%282%288000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18500 ALIGN=MIDDLE  ALT=\"18500\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=342250000 ALIGN=MIDDLE  ALT=\"342250000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+sqrt%28+342250000--30240000+%29%29%2F%282%288000%29%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+sqrt%28+342250000--30240000+%29%29%2F%282%288000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%288000%29%28-945%29 ALIGN=MIDDLE  ALT=\"4%288000%29%28-945%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30240000 ALIGN=MIDDLE  ALT=\"-30240000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+sqrt%28+342250000%2B30240000+%29%29%2F%282%288000%29%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+sqrt%28+342250000%2B30240000+%29%29%2F%282%288000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28342250000--30240000%29 ALIGN=MIDDLE  ALT=\"sqrt%28342250000--30240000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28342250000%2B30240000%29 ALIGN=MIDDLE  ALT=\"sqrt%28342250000%2B30240000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%282%288000%29%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%282%288000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=342250000 ALIGN=MIDDLE  ALT=\"342250000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30240000 ALIGN=MIDDLE  ALT=\"30240000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=372490000 ALIGN=MIDDLE  ALT=\"372490000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%2816000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8000 ALIGN=MIDDLE  ALT=\"8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16000 ALIGN=MIDDLE  ALT=\"16000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B-+19300%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B-+19300%29%2F%2816000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=372490000 ALIGN=MIDDLE  ALT=\"372490000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19300 ALIGN=MIDDLE  ALT=\"19300\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+%2B+19300%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+%2B+19300%29%2F%2816000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-18500+-+19300%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=+%28-18500+-+19300%29%2F%2816000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28800%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=+%28800%29%2F%2816000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=++%28-37800%29%2F%2816000%29 ALIGN=MIDDLE  ALT=\"r+=++%28-37800%29%2F%2816000%29\"   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=r+=+0.05 ALIGN=MIDDLE  ALT=\"r+=+0.05\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-2.3625 ALIGN=MIDDLE  ALT=\"r+=+-2.3625\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide \r<BR>\n" );
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document.write( "So the <i>possible</i> answers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+0.05 ALIGN=MIDDLE  ALT=\"r+=+0.05\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-2.3625 ALIGN=MIDDLE  ALT=\"r+=+-2.3625\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  <BR>\n" );
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document.write( "However, since you CANNOT have a negative interest rate, this means that we can ignore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-2.3625 ALIGN=MIDDLE  ALT=\"r+=+-2.3625\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So this means that the only answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+0.05 ALIGN=MIDDLE  ALT=\"r+=+0.05\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now multiply by 100 (to convert to a percentage) to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2A0.05=5 ALIGN=MIDDLE  ALT=\"100%2A0.05=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the annual interest rate is 5%\r<BR>\n" );
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document.write( "That's quite a guess :) </SPAN>\n" );
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