document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207173>Question 1207173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question content area top<BR>\n" );
document.write( "Part 1<BR>\n" );
document.write( "A person invested ​$6900 for 1​ year, part at 8​%, part at 11​%, and the remainder at 14​%. The total annual income from these investments was ​$834. The amount of money invested at 14% was ​$900 more than the amounts invested at 8​% and 11​% combined. Find the amount invested at each rate.<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 #844915 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There are always different ways to set up a problem for solving.<br><BR>\n" );
document.write( "The method shown by the other tutor basically translates the given information directly into equations that are then solved to find the answer.<br><BR>\n" );
document.write( "I think a very different approach, analyzing the given information to find a way to set up the problem that makes solving the problem easier, is better.<br><BR>\n" );
document.write( "Given that the total invested was $6900 and that the amount invested at 14% was $900 more than the combined amounts invested at 8% and 11%, we can quickly determine that $3900 was invested at 14% and $3000 combined at 8% and 11%.  I leave the details for finding that result to the student, if formal algebra is required.<br><BR>\n" );
document.write( "$3900 invested at 14% yields $546 income; that leaves $834-$546 = $288 income from the other two investments.<br><BR>\n" );
document.write( "A standard formal solution for that part of the problem would start something like this:<br><BR>\n" );
document.write( "x = amount invested at 8%<BR>\n" );
document.write( "3000-x = amount invested at 11%<br><BR>\n" );
document.write( "The total income from those two investments was $288:<br><BR>\n" );
document.write( ".08(x)+.11(3000-x)=288<br><BR>\n" );
document.write( "Solving that equation then finishes the problem.<br><BR>\n" );
document.write( "I use a different, less formal method for solving 2-part \"mixture\" problems like this.  Here is my method for solving this part of the problem.<br><BR>\n" );
document.write( "All $3000 invested at 8% would yield $240 income; all invested at 11% would yield $330 income; the actual income from those two investments was $288.<BR>\n" );
document.write( "Look at those three numbers 240, 288, and 330 (on a number line, if it helps), and observe/calculate that 288 is 48/90 = 8/15 of the way from 240 to 330.<BR>\n" );
document.write( "That means 8/15 of the $3000 was invested at the higher rate.<br><BR>\n" );
document.write( "8/15 of $3000 is 8*$200 = $1600, so $1600 was invested at 11% and $1400 at 8%.<br><BR>\n" );
document.write( "ANSWERS: $3900 at 14%, $1600 at 11%, and $1400 at 8%<br><BR>\n" );
document.write( "CHECK: .14(3900)+.11(1600)+.08(1400) = 546+176+112 = 834<br><BR>\n" );
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