document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83291>Question 83291</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I get so lost in these long problems please help\r<BR>\n" );
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document.write( "The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of Deposit) is given by<BR>\n" );
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document.write( "A is the amount of returned.<BR>\n" );
document.write( "P is the principal amount initially deposited.<BR>\n" );
document.write( "r is the annual interest rate (expressed as a decimal).<BR>\n" );
document.write( "n is the compound period.<BR>\n" );
document.write( "t is the number of years.\r<BR>\n" );
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document.write( "Carry all calculations to 6 decimals on all assignments then round the answer to the nearest cent.\r<BR>\n" );
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document.write( "Suppose you deposit $10,000 for 2 years at a rate of 10%.\r<BR>\n" );
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document.write( "a) 	Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "Answer:	<BR>\n" );
document.write( "a) continued<BR>\n" );
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document.write( "b) 	Calculate the return (A) if the bank compounds quarterly (n = 4).  Round your answer to the hundredth's place.<BR>\n" );
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document.write( "c) 	Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "d) 	Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "Answer:	\r<BR>\n" );
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document.write( "e)	What observation can you make about the size of the increase in your return as your compounding increases more frequently? <BR>\n" );
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document.write( "f) 	If a bank compounds continuously, then the formula takes a simpler, that is <BR>\n" );
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document.write( "where e is a constant and equals approximately 2.7183.<BR>\n" );
document.write( "Calculate A with continuous compounding. Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "Answer:	\r<BR>\n" );
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document.write( "g) 	Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "Answer: 	\r<BR>\n" );
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document.write( "h) 	A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #59888 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=59888\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Using this formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2Br%2Fn%29%5E%28nt%29 ALIGN=MIDDLE  ALT=\"A=P%281%2Br%2Fn%29%5E%28nt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > calculate the following:\r<BR>\n" );
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document.write( "a)Calculate the return (A) if the bank compounds annually (n = 1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.10%2F1%29%5E%281%2A2%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.10%2F1%29%5E%281%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Start with the given expression<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.1%29%5E%282%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.1%29%5E%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     Divide 0.10 by 1 and multiply the exponents 1 and 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.1%29%5E%282%29 ALIGN=MIDDLE  ALT=\"A=10000%281.1%29%5E%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Add<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.21%29 ALIGN=MIDDLE  ALT=\"A=10000%281.21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >      Raise 1.1 to 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=12100 ALIGN=MIDDLE  ALT=\"A=12100\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Multiply\r<BR>\n" );
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document.write( "So the return is $12,100\r<BR>\n" );
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document.write( "b)Calculate the return (A) if the bank compounds quarterly (n = 4)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.10%2F4%29%5E%284%2A2%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.10%2F4%29%5E%284%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Start with the given expression<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.025%29%5E%288%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.025%29%5E%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     Divide 0.10 by 4 and multiply the exponents 4 and 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.025%29%5E%288%29 ALIGN=MIDDLE  ALT=\"A=10000%281.025%29%5E%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Add<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.21840289750992%29 ALIGN=MIDDLE  ALT=\"A=10000%281.21840289750992%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >      Raise 1.025 to 8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=12184.0289750992 ALIGN=MIDDLE  ALT=\"A=12184.0289750992\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Multiply\r<BR>\n" );
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document.write( "So the return is $12184.03\r<BR>\n" );
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document.write( "c)Calculate the return (A) if the bank compounds monthly(n = 12)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.10%2F12%29%5E%2812%2A2%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.10%2F12%29%5E%2812%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Start with the given expression<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.00833333333333333%29%5E%2824%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.00833333333333333%29%5E%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     Divide 0.10 by 12 and multiply the exponents 12 and 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.00833333333333%29%5E%2824%29 ALIGN=MIDDLE  ALT=\"A=10000%281.00833333333333%29%5E%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Add<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.22039096137556%29 ALIGN=MIDDLE  ALT=\"A=10000%281.22039096137556%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >      Raise 1.00833333333333 to 24<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=12203.9096137556 ALIGN=MIDDLE  ALT=\"A=12203.9096137556\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Multiply\r<BR>\n" );
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document.write( "So the return is $12203.91\r<BR>\n" );
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document.write( "d)Calculate the return (A) if the bank compounds daily (n = 365)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.10%2F365%29%5E%28365%2A2%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.10%2F365%29%5E%28365%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Start with the given expression<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281%2B0.000273972602739726%29%5E%28730%29 ALIGN=MIDDLE  ALT=\"A=10000%281%2B0.000273972602739726%29%5E%28730%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     Divide 0.10 by 365 and multiply the exponents 365 and 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.00027397260274%29%5E%28730%29 ALIGN=MIDDLE  ALT=\"A=10000%281.00027397260274%29%5E%28730%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Add<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.22136930163979%29 ALIGN=MIDDLE  ALT=\"A=10000%281.22136930163979%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >      Raise 1.00027397260274 to 730<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=12213.6930163979 ALIGN=MIDDLE  ALT=\"A=12213.6930163979\"   DISABLED_style= \"max-width:90%; height:auto;\" >       Multiply\r<BR>\n" );
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document.write( "So the return is $12213.69\r<BR>\n" );
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document.write( "e) What observation can you make about the size of the increase in your return as your compounding increases more frequently?\r<BR>\n" );
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document.write( "As the compounding frequency increases, the return slowly approaches some finite number (which in this case appears to be about $12213.69). Think about it, banks wouldn't be too fond of shelling out an infinite amount of cash.\r<BR>\n" );
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document.write( "f)Calculate A with continuous compounding\r<BR>\n" );
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document.write( "Using the contiuous compounding formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=Pe%5E%28rt%29 ALIGN=MIDDLE  ALT=\"A=Pe%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where e is the constant 2.7183 and letting r=0.1, P=10,000, and t=2 we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%282.7183%29%5E%280.1%2A2%29 ALIGN=MIDDLE  ALT=\"A=10000%282.7183%29%5E%280.1%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%282.7183%29%5E%280.2%29 ALIGN=MIDDLE  ALT=\"A=10000%282.7183%29%5E%280.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 0.1 and 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10000%281.22140439115573%29 ALIGN=MIDDLE  ALT=\"A=10000%281.22140439115573%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise 2.7183 to 0.2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=12214.0439115572 ALIGN=MIDDLE  ALT=\"A=12214.0439115572\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "So using continuous compounding interest we get a return of $12,214.04 (which is real close to what we got from a daily compounding frequency)\r<BR>\n" );
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document.write( "g)Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000%2F10000=e%5E%280.1t%29 ALIGN=MIDDLE  ALT=\"15000%2F10000=e%5E%280.1t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 10,000\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5=e%5E%280.1t%29 ALIGN=MIDDLE  ALT=\"1.5=e%5E%280.1t%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=ln%281.5%29=0.1t ALIGN=MIDDLE  ALT=\"ln%281.5%29=0.1t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of both sides. This eliminates \"e\".The natural log (pronounced \"el\" \"n\") is denoted \"ln\" on calculators.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281.5%29%2F0.1=t ALIGN=MIDDLE  ALT=\"ln%281.5%29%2F0.1=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 0.1\r<BR>\n" );
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document.write( "So we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4.054 ALIGN=MIDDLE  ALT=\"t=4.054\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So it will take about 4 years to generate $15,000\r<BR>\n" );
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document.write( "h) A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer?\r<BR>\n" );
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document.write( "Since we want to double our money, let A=2*10,000. So A=20,000. Now solve for t:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20000=10000e%5E%280.1t%29 ALIGN=MIDDLE  ALT=\"20000=10000e%5E%280.1t%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=20000%2F10000=e%5E%280.1t%29 ALIGN=MIDDLE  ALT=\"20000%2F10000=e%5E%280.1t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 10,000\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=e%5E%280.1t%29 ALIGN=MIDDLE  ALT=\"2=e%5E%280.1t%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=ln%282%29=0.1t ALIGN=MIDDLE  ALT=\"ln%282%29=0.1t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of both sides. This eliminates \"e\".The natural log (pronounced \"el\" \"n\") is denoted \"ln\" on calculators.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29%2F0.1=t ALIGN=MIDDLE  ALT=\"ln%282%29%2F0.1=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 0.1\r<BR>\n" );
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document.write( "So we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=6.9314 ALIGN=MIDDLE  ALT=\"t=6.9314\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So it will take about 7 years to double your money.\r<BR>\n" );
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