document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=566374>Question 566374</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You have saved a total of more than $75 but less than $120.  When you count your money, you find that you have some one dollar bills, some five, and some tens.  You have twice as many tens as fives and four times as many ones than fives.  How much money could you have?  Write equations and/or inequalities (using \"t\" for ttens, \"f\" for fives and \"n\" for ones) in order to solve.  Explain your work.\r<BR>\n" );
document.write( "\n" );
document.write( "I know I have to start with 2t than 5 + 4x than ones.  Now I am stuck. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #366217 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=366217\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the number of $1 bills;<BR>\n" );
document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the number of $5 bills, and<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the number of $10 bills, as suggested<BR>\n" );
document.write( "We start with equations, because some numbers seem to be equal.<BR>\n" );
document.write( "We can translate \"twice as many tens as fives\" as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2f ALIGN=MIDDLE  ALT=\"t=2f\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can translate \"four times as many ones than fives\" as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=4f ALIGN=MIDDLE  ALT=\"n=4f\"   DISABLED_style= \"max-width:90%; height:auto;\" > , although I would have stated that as \"four times as many ones as fives.\" <BR>\n" );
document.write( "The total amount of money would be $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10t%2B5f%2Bn ALIGN=MIDDLE  ALT=\"10t%2B5f%2Bn\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but we can substitute the expressions above to have everything as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the number of $5 bills.<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10t%2B5f%2Bn ALIGN=MIDDLE  ALT=\"10t%2B5f%2Bn\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%282f%29%2B5f%2B%284f%29 ALIGN=MIDDLE  ALT=\"10%282f%29%2B5f%2B%284f%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20f%2B5f%2B4f ALIGN=MIDDLE  ALT=\"20f%2B5f%2B4f\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29f ALIGN=MIDDLE  ALT=\"29f\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we have $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29f ALIGN=MIDDLE  ALT=\"29f\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We know that the amount is \"more than $75 but less than $120,\" so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=75%3C29f%3C120 ALIGN=MIDDLE  ALT=\"75%3C29f%3C120\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--- That's two inequalities in one, but it's OK to write it like this (at least in my book).<BR>\n" );
document.write( "So we divide all 3 sides of the inequalities by 29 and get an equivalent double inequality.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=75%2F29%3Cf%3C120%2F29 ALIGN=MIDDLE  ALT=\"75%2F29%3Cf%3C120%2F29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=75%2F29 ALIGN=MIDDLE  ALT=\"75%2F29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is approximately 2.586 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%2F29 ALIGN=MIDDLE  ALT=\"120%2F29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is approximately 4.138, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > is suposed to be a natural number (a positive integer).<BR>\n" );
document.write( "So it could only be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=3 ALIGN=MIDDLE  ALT=\"f=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=4 ALIGN=MIDDLE  ALT=\"f=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=3 ALIGN=MIDDLE  ALT=\"f=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the amount of money you have is $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29f ALIGN=MIDDLE  ALT=\"29f\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29%2A3 ALIGN=MIDDLE  ALT=\"29%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$87.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=4 ALIGN=MIDDLE  ALT=\"f=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the amount of money you have is $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29f ALIGN=MIDDLE  ALT=\"29f\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29%2A4 ALIGN=MIDDLE  ALT=\"29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$116. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );