document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200766>Question 1200766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A loan of $45,000 is made at 5.5% interest, compounded annually. After how many years will the amount due reach $70,000 or more? Write the smallest possible whole number answer. </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>9 years</font>\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "The compound interest formula we use here is<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)\r<BR>\n" );
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document.write( "In this case,<BR>\n" );
document.write( "A = 70000 = final amount<BR>\n" );
document.write( "P = 45000 = starting amount<BR>\n" );
document.write( "r = 0.055 = decimal form of the 5.5% interest rate<BR>\n" );
document.write( "n = 1 = since we're compounding once per year<BR>\n" );
document.write( "t = unknown number of years.\r<BR>\n" );
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document.write( "We use logarithms to isolate the exponent.<BR>\n" );
document.write( "Think of it like saying: \"The variable is in the trees so we must log it down.\"<BR>\n" );
document.write( "The reason why logs are handy here is that we'll take advantage of the rule log(A^B) = B*log(A) to pull down the exponent.\r<BR>\n" );
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document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "70000 = 45000*(1+0.055/1)^(1*t)<BR>\n" );
document.write( "70000 = 45000*(1.055)^t<BR>\n" );
document.write( "70000/45000 = (1.055)^t<BR>\n" );
document.write( "1.555556 = (1.055)^t<BR>\n" );
document.write( "log( 1.555556 ) = log( (1.055)^t )<BR>\n" );
document.write( "log( 1.555556 ) = t*log( 1.055 ) .... use the log rule mentioned<BR>\n" );
document.write( "t*log( 1.055 ) = log( 1.555556 )<BR>\n" );
document.write( "t = log(1.555556)/log(1.055)<BR>\n" );
document.write( "t = 8.252273<BR>\n" );
document.write( "t = 9\r<BR>\n" );
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document.write( "We round UP to the nearest integer.<BR>\n" );
document.write( "It doesn't matter that 8.252273 is closer to 8 than it is to 9.\r<BR>\n" );
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document.write( "If we tried t = 8, then,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = 45000*(1+0.055/1)^(1*8)<BR>\n" );
document.write( "A = 69060.8931748269<BR>\n" );
document.write( "A = 69060.89<BR>\n" );
document.write( "which is about $1000 short of the goal of $70,000.\r<BR>\n" );
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document.write( "But t = 9 leads to,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = 45000*(1+0.055/1)^(1*9)<BR>\n" );
document.write( "A = 72859.2422994423<BR>\n" );
document.write( "A = 72859.24<BR>\n" );
document.write( "we've satisfied the \"$70,000 or more\" criteria.<BR>\n" );
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