document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1100944>Question 1100944</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The repair department of the bicycle shop repairs three things: flat tires, bent handle bars and<BR>\n" );
document.write( "ripped seats. Today in the repair department, 25% of the bikes had flat tires only, 5% had bent<BR>\n" );
document.write( "handlebars only, and 10% had ripped seats only. Just 1/12th of the bikes had all three repairs to<BR>\n" );
document.write( "do: flat tires, bent handlebars and ripped seats. No bikes were completely fixed and there are a<BR>\n" );
document.write( "total of 101 repairs to be made. How many bikes are in the repair department? How many bikes<BR>\n" );
document.write( "need two repairs? If less than half of all the bikes have a ripped seat, what is the range of bikes<BR>\n" );
document.write( "that need both the tires and handlebars repaired without needing to fix the seat? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #715526 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=715526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ONE APPROACH:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of bikes in the shop waiting to be repaired.<BR>\n" );
document.write( "Of those,<BR>\n" );
document.write( "25% have flat tires only,<BR>\n" );
document.write( "5% have bent handlebars only, and<BR>\n" );
document.write( "10% have ripped seats only.<BR>\n" );
document.write( "That adds up to 25%+5%+10%=40% .<BR>\n" );
document.write( "So, the number of bikes needing only 1 repair is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2840%2F100%29N=%282%2F5%29N ALIGN=MIDDLE  ALT=\"%2840%2F100%29N=%282%2F5%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The number of bikes needing 3 repairs is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29N ALIGN=MIDDLE  ALT=\"%281%2F12%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The rest of the bikes is the fraction of all bikes that need exactly 2 repairs.<BR>\n" );
document.write( "As a fraction of all the bikes, that is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2%2F5-1%2F12=60%2F60-24%2F60-5%2F60=31%2F60 ALIGN=MIDDLE  ALT=\"1-2%2F5-1%2F12=60%2F60-24%2F60-5%2F60=31%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As a number of bikes, it is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N-%282%2F5%29N-%281%2F12%29N=%281-2%2F5%2F1%2F12%29N=%2831%2F60%29N ALIGN=MIDDLE  ALT=\"N-%282%2F5%29N-%281%2F12%29N=%281-2%2F5%2F1%2F12%29N=%2831%2F60%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Counting the number of repairs needed, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29N ALIGN=MIDDLE  ALT=\"%282%2F5%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > repairs for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29N ALIGN=MIDDLE  ALT=\"%282%2F5%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that need only 1 repair,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2831%2F60%29N=%2831%2F30%29N ALIGN=MIDDLE  ALT=\"2%2831%2F60%29N=%2831%2F30%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > repairs for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2831%2F60%29N ALIGN=MIDDLE  ALT=\"%2831%2F60%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that need exactly 2 repairs,<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%281%2F12%29N=%281%2F4%29N ALIGN=MIDDLE  ALT=\"3%281%2F12%29N=%281%2F4%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > repairs for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29N ALIGN=MIDDLE  ALT=\"%281%2F12%29N\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that need 3 repairs.<BR>\n" );
document.write( "The number of repairs adds up to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29N%2B%2831%2F60%29N%2B%281%2F4%29N=101 ALIGN=MIDDLE  ALT=\"%282%2F5%29N%2B%2831%2F60%29N%2B%281%2F4%29N=101\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%2B31%2F60%2B1%2F4%29N=101 ALIGN=MIDDLE  ALT=\"%282%2F5%2B31%2F60%2B1%2F4%29N=101\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2824%2F60%2B31%2F60%2B15%2F60%29N=101 ALIGN=MIDDLE  ALT=\"%2824%2F60%2B31%2F60%2B15%2F60%29N=101\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28101%2F60%29N=101 ALIGN=MIDDLE  ALT=\"%28101%2F60%29N=101\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2860%2F101%29%28101%2F60%29N=%2830%2F101%29101 ALIGN=MIDDLE  ALT=\"%2860%2F101%29%28101%2F60%29N=%2830%2F101%29101\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28N=60%29 ALIGN=MIDDLE  ALT=\"highlight%28N=60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2860%29 ALIGN=MIDDLE  ALT=\"highlight%2860%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes in the repair department.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2831%2F60%2960=highlight%2831%29 ALIGN=MIDDLE  ALT=\"%2831%2F60%2960=highlight%2831%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > need exactly two repairs,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F12 ALIGN=MIDDLE  ALT=\"1%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes, need all 3 repairs including their ripped seats,<BR>\n" );
document.write( "25% , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes, only need flat tires fixed,<BR>\n" );
document.write( "5% , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes, only need bent handlebars fixed, and<BR>\n" );
document.write( "10%, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes only need their ripped seats fixed.<BR>\n" );
document.write( "In pictures, we could represent is as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-2.6%2C2.6%2C-1.75%2C3.45%2C%0D%0Ared%28circle%280%2C1.732%2C1.49%29%29%2Cred%28circle%280%2C1.732%2C1.51%29%29%2C%0D%0Ared%28arrow%282%2C3.031%2C0.75%2C3.031%29%29%2Clocate%282.05%2C3.15%2Cred%28R%29%29%2C%0D%0Alocate%281.88%2C2.97%2Cred%28%22%28+ripped%22%29%29%2Clocate%281.9%2C2.75%2Cred%28%22seats+%29%22%29%29%2C%0D%0Alocate%280%2C2%2C6%29%2Clocate%280%2C0.77%2C5%29%2C%0D%0Acircle%281%2C0%2C1.5%29%2Ccircle%28-1%2C0%2C1.5%29%2C%0D%0Alocate%28-1.2%2C0%2C15%29%2Clocate%281.2%2C0%2C3%29%2C%0D%0Aarrow%28-2.06%2C1.9%2C-2.06%2C1.06%29%2Clocate%28-2.3%2C2.1%2Ctires%29%2C%0D%0Aarrow%282.06%2C1.9%2C2.06%2C1.06%29%2Clocate%281.7%2C2.1%2Chandlebars%29%2C%0D%0Alocate%28-0.8%2C1%2C%22%3F%22%29%2Clocate%280.75%2C1%2C%22%3F%22%29%2Clocate%280%2C-0.1%2Cx%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( "The last question asks for the range of values for value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "The number of \"bikes<BR>\n" );
document.write( "that need both the tires and handlebars repaired without needing to fix the seat\".<BR>\n" );
document.write( " <BR>\n" );
document.write( "From the picture,<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%3F%22%2B%22%3F%22=0 ALIGN=MIDDLE  ALT=\"%22%3F%22%2B%22%3F%22=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60-6-5-15-3=31 ALIGN=MIDDLE  ALT=\"x=60-6-5-15-3=31\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the greatest possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ;<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B5%2B%22%3F%22%2B%22%3F%22=29 ALIGN=MIDDLE  ALT=\"6%2B5%2B%22%3F%22%2B%22%3F%22=29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60-29-15-3=13 ALIGN=MIDDLE  ALT=\"x=60-29-15-3=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the least possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Without a picture:<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the total number of ripped seat repairs,<BR>\n" );
document.write( "we do not know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > exactly, but<BR>\n" );
document.write( "it includes at least the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > that need only the seat repaired,<BR>\n" );
document.write( "and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > that need all 3 repairs, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%3E=6%2B5 ALIGN=MIDDLE  ALT=\"R%3E=6%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%3C=R ALIGN=MIDDLE  ALT=\"11%3C=R\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If less than half the bikes have a ripped seat<BR>\n" );
document.write( "(meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29 ALIGN=MIDDLE  ALT=\"29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or less bikes),<BR>\n" );
document.write( "we also know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%3C=29 ALIGN=MIDDLE  ALT=\"R%3C=29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%3C=R%3C=29 ALIGN=MIDDLE  ALT=\"11%3C=R%3C=29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If we exclude <BR>\n" );
document.write( "those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes with ripped seats,<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that only need flat tires fixed,<BR>\n" );
document.write( "and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that only need bent handlebars fixed,<BR>\n" );
document.write( "we are excluding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2B15%2B3=R%2B18 ALIGN=MIDDLE  ALT=\"R%2B15%2B3=R%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "what is left is the group of bikes that needs<BR>\n" );
document.write( "\"both the tires and handlebars repaired without needing to fix the seat.\"<BR>\n" );
document.write( "That number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60-%28R%2B18%29=60-R-18=42-R ALIGN=MIDDLE  ALT=\"x=60-%28R%2B18%29=60-R-18=42-R\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%3C=R%3C=29 ALIGN=MIDDLE  ALT=\"11%3C=R%3C=29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42-11%3E=42-R%3E=42-29 ALIGN=MIDDLE  ALT=\"42-11%3E=42-R%3E=42-29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31%3E=x%3E=13 ALIGN=MIDDLE  ALT=\"31%3E=x%3E=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are at least <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > and at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31 ALIGN=MIDDLE  ALT=\"31\"   DISABLED_style= \"max-width:90%; height:auto;\" > bikes that need <BR>\n" );
document.write( "\"both the tires and handlebars repaired without needing to fix the seat.\"<BR>\n" );
document.write( "The range can be expressed as [13,31], or as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%3C=x%3C=31 ALIGN=MIDDLE  ALT=\"13%3C=x%3C=31\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );