document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176067>Question 1176067</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A state lotto has a prize that pays $1,700 each week for 40 years.\r<BR>\n" );
document.write( "\n" );
document.write( "Find the total value of the prize: $<BR>\n" );
document.write( "Correct  \r<BR>\n" );
document.write( "\n" );
document.write( "If the state can earn 3% interest on investments, how much money will they need to put into an account now to cover the weekly prize payments? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850585 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850585\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's calculate the total value of the prize and the amount needed to cover the weekly payments.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Total Value of the Prize**\r<BR>\n" );
document.write( "\n" );
document.write( "* Weekly prize: $1,700<BR>\n" );
document.write( "* Years: 40<BR>\n" );
document.write( "* Weeks per year: 52\r<BR>\n" );
document.write( "\n" );
document.write( "Total value = Weekly prize * Weeks per year * Years<BR>\n" );
document.write( "Total value = $1,700 * 52 * 40<BR>\n" );
document.write( "Total value = $3,536,000\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Present Value Calculation**\r<BR>\n" );
document.write( "\n" );
document.write( "To determine how much money the state needs to invest now, we need to calculate the present value of the annuity.\r<BR>\n" );
document.write( "\n" );
document.write( "* Weekly payment: $1,700<BR>\n" );
document.write( "* Interest rate: 3% per year<BR>\n" );
document.write( "* Years: 40\r<BR>\n" );
document.write( "\n" );
document.write( "Since the payments are weekly, we need to adjust the interest rate to a weekly rate.\r<BR>\n" );
document.write( "\n" );
document.write( "* Annual interest rate: 0.03<BR>\n" );
document.write( "* Weekly interest rate: 0.03 / 52 ≈ 0.000576923\r<BR>\n" );
document.write( "\n" );
document.write( "We'll use the present value of an ordinary annuity formula, but since it is weekly, we will use the weekly interest rate, and total number of weeks.\r<BR>\n" );
document.write( "\n" );
document.write( "Total number of weeks = 40 years * 52 weeks/year = 2080 weeks\r<BR>\n" );
document.write( "\n" );
document.write( "Using the present value of an annuity formula:\r<BR>\n" );
document.write( "\n" );
document.write( "PV = PMT * [1 - (1 + r)^-n] / r\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* PV = Present Value<BR>\n" );
document.write( "* PMT = Weekly payment ($1,700)<BR>\n" );
document.write( "* r = Weekly interest rate (0.03 / 52)<BR>\n" );
document.write( "* n = Total number of weeks (2080)\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 1700 * [1 - (1 + 0.03/52)^-2080] / (0.03/52)\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 1700 * [1 - (1.000576923)^-2080] / 0.000576923\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 1700 * [1 - 0.301131] / 0.000576923\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 1700 * 0.698869 / 0.000576923\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 1700 * 1211.37\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 2059329\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the state needs to put approximately $2,059,329 into an account now to cover the weekly prize payments.\r<BR>\n" );
document.write( "\n" );
document.write( "**Answers:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Total value of the prize: $3,536,000<BR>\n" );
document.write( "* Amount needed to cover the prize payments: $2,059,329 (approximately)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );