document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1034821>Question 1034821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Nicole invested a certain amount of money for one year and earned $78 in interest. Jon invested $260 more at an interest rate that was 1% less than Nicole, but earned the same amount of interest. Find Nicole's principal and interest rate. \r<BR>\n" );
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document.write( "please explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #649559 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=649559\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the amount she invested.<BR>\n" );
document.write( "let r = the rate of interest she invested at.<BR>\n" );
document.write( "since her interest earned was 78, her equation becomes:<BR>\n" );
document.write( "x*r = 78\r<BR>\n" );
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document.write( "jon invested 260 more at an interest rate that was 1% less.<BR>\n" );
document.write( "1% is equal to .1<BR>\n" );
document.write( "jon's equation becomes:<BR>\n" );
document.write( "(x+260) * (r-.1) = 78\r<BR>\n" );
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document.write( "you have 2 equations that need to be solved simultaneously.<BR>\n" );
document.write( "they are:\r<BR>\n" );
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document.write( "x*r = 78<BR>\n" );
document.write( "(x+260) * (r-.1) = 78\r<BR>\n" );
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document.write( "this system can be solved in a number of ways.<BR>\n" );
document.write( "one way to solve it is as follows.\r<BR>\n" );
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document.write( "since both expressions on the left side of each equation are equal to 78, you can those expressions equal to each other to get:\r<BR>\n" );
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document.write( "x*r = (x+260) * (r-.1)\r<BR>\n" );
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document.write( "if you expand this equation, you get:\r<BR>\n" );
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document.write( "x*r = x*r - .1*x + 260*r - .1*260\r<BR>\n" );
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document.write( "simplify this further to get:\r<BR>\n" );
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document.write( "x*r = x*r - .1*x + 260*r - 26\r<BR>\n" );
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document.write( "if you subtract x*r from both sides of the equation, they cancel out and you are left with:\r<BR>\n" );
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document.write( "0 = -.1*x + 260*r - 26\r<BR>\n" );
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document.write( "this is the same as -.1*x + 260*r - 26 = 0\r<BR>\n" );
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document.write( "you want to reduce the number of variables in one equation to one so it can be solved.\r<BR>\n" );
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document.write( "go back to your original equations.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "x*r = 78<BR>\n" );
document.write( "(x+260) * (r-.1) = 78\r<BR>\n" );
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document.write( "from the first of these equations, solve for x to get x = 78/r\r<BR>\n" );
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document.write( "in your equation of -.1*x + 260*r - 26 = 0, replace x with 78/r to get:\r<BR>\n" );
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document.write( "-.1*78/r + 260*r - 26 = 0\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "-7.8/r + 260*r - 26 = 0\r<BR>\n" );
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document.write( "multiply both sides of this equation by r to get:\r<BR>\n" );
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document.write( "-7.8 + 260*r^2 - 26*r = 0\r<BR>\n" );
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document.write( "order the terms of this equation in descending order of degree to get:\r<BR>\n" );
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document.write( "260*r^2 - 26*r - 7.8 = 0\r<BR>\n" );
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document.write( "use the quadratic formula to solve this to get:\r<BR>\n" );
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document.write( "r = 0.2302775637732 or x = -0.1302775637732\r<BR>\n" );
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document.write( "your calculator might not give you as many as these digits, but mine did so i used what i had.  if your calculator gives you less, then use that.   most calculator will take you out to about 8 or 9 digits accuracy.  this calculator was an online one that gave me more.\r<BR>\n" );
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document.write( "it's not possible to get a positive interest if the interest rate is negative, so the solution has to be r = 0.2302775637732\r<BR>\n" );
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document.write( "since r = 0.2302775637732, then (r-.1) must be equal to 0.1302775637732\r<BR>\n" );
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document.write( "from the first equation of x*r = 78, you get:<BR>\n" );
document.write( "x * 0.2302775637732 = 78<BR>\n" );
document.write( "solve for x to get x = 78 / 0.2302775637732 = 338.72166581<BR>\n" );
document.write( "that's nicole's principal.\r<BR>\n" );
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document.write( "x+260 = 598.72166581<BR>\n" );
document.write( "598.72166581 * 0.1302775637732 = 78<BR>\n" );
document.write( "that's jon's principal multiplied by jon's interest rate taken from the formula:<BR>\n" );
document.write( "(x+260) * (r-.1) = 78.<BR>\n" );
document.write( "this calculation confirms that nicole's principal and nicole's interest rate are correct because joh's principal is nocile's principal plus 260 and jon's interest rate is nicole's interest rate - .1.\r<BR>\n" );
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document.write( "your solution is:\r<BR>\n" );
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document.write( "nicole's principal is 338.72166581 dollars.<BR>\n" );
document.write( "nicole's interest rate is 0.2302775637732\r<BR>\n" );
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document.write( "lots of digits here to maintain the accuracy.<BR>\n" );
document.write( "the concept is fairly simple.<BR>\n" );
document.write( "the execution is a bit complicated.\r<BR>\n" );
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document.write( "the concept is as stated above.<BR>\n" );
document.write( "x * r = 78<BR>\n" );
document.write( "(x + 260) * (r - .1) = 78\r<BR>\n" );
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document.write( "i set the expressions on the left side of each of those equaitons equal to each other<BR>\n" );
document.write( "that's one way to solve it, but you could also have solved by the normal method of substitution.\r<BR>\n" );
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document.write( "the equations are still:<BR>\n" );
document.write( "x * r = 78<BR>\n" );
document.write( "(x + 260) * (r - .1) = 78\r<BR>\n" );
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document.write( "in the first equation, solve for x to get x = 78/r\r<BR>\n" );
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document.write( "simplify the second equation to get:\r<BR>\n" );
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document.write( "x*r - .1*x + 260*r - .1*260 = 78\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "x*r - .1*x + 260*r - 26 = 78\r<BR>\n" );
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document.write( "replace x in the second equation with 78/r derived from the first equation to get:\r<BR>\n" );
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document.write( "78/r*r - .1*78/r + 260*r - 26 = 78\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "78 - 7.8/r + 260*r - 26 = 78\r<BR>\n" );
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document.write( "multiply both sides of the equation by r to get:\r<BR>\n" );
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document.write( "78*r - 7.8 + 260*r^2 - 26*r = 78*r\r<BR>\n" );
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document.write( "subtract 78*r from both sides of the equation to get:\r<BR>\n" );
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document.write( "-7.8 + 260*r^2 - 26*r = 0\r<BR>\n" );
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document.write( "reorder terms in descending order of degree to get:\r<BR>\n" );
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document.write( "260*r^2 - 26*r - 7.8 = 0\r<BR>\n" );
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document.write( "this is exactly the same equation we got when we used the other method.<BR>\n" );
document.write( "you get the same answer as you should regardless of the method you choose as long as you execute that method correctly.\r<BR>\n" );
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document.write( "doing this manually is a chore.<BR>\n" );
document.write( "using a calculator helps.<BR>\n" );
document.write( "being able to store intermediate results in memory and then reusing those results later on reduces the manual burden involved.<BR>\n" );
document.write( "you should not round until you get to the final result, otherwise you can lose accuracy.<BR>\n" );
document.write( "hopefully this helps you understand the concept and a couple of ways to execute on the methods involved.<BR>\n" );
document.write( "don't sweat the large number of decimal digits i used.<BR>\n" );
document.write( "it's because the calculator i used allowed that.<BR>\n" );
document.write( "your calculator will more then likely display less.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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