document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=713489>Question 713489</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that during a test drive of two cars, one car travels 196 miles in the same time that the second car travels 140 miles. If the speed of one car is 16 miles per hour faster than the speed of the second car, find the speed of both cars. \r<BR>\n" );
document.write( "\n" );
document.write( "The speed of the first car is __ mph, and the speed of the second car is __ mph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #438495 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=438495\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I can think of many ways to solve the problem that make sense to me.<BR>\n" );
document.write( "With some luck, at least one of those ways will make sense to you.<BR>\n" );
document.write( "If you are expected to show and explain your work, there may be one way that is expected, or preferred.<BR>\n" );
document.write( "In that case, I hope that the teacher understands that any way to solve a problem is a good way to solve it.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE FIFTH GRADER WAY:<BR>\n" );
document.write( "An extra 16 mph (miles per hour) resulted in an extra 56 miles (196-140=156) for the first car.<BR>\n" );
document.write( "Then the time for that long test drive (in hours) must have been<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56%2F16=3.5 ALIGN=MIDDLE  ALT=\"56%2F16=3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The test drive took 3.5 hours.<BR>\n" );
document.write( "The speed of the first car (in mph) was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196%2F3.5 ALIGN=MIDDLE  ALT=\"196%2F3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour.<BR>\n" );
document.write( "The speed of the second car (in mph) was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140%2F3.5 ALIGN=MIDDLE  ALT=\"140%2F3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour.<BR>\n" );
document.write( "With a calculator those division are easy to do.<BR>\n" );
document.write( "Without a calculator, I can calculate it easier by first multiplying numerator and denominator times 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196%2F3.5=196%2A2%2F3.5%2F2=392%2F7=highlight%2856%29 ALIGN=MIDDLE  ALT=\"196%2F3.5=196%2A2%2F3.5%2F2=392%2F7=highlight%2856%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140%2F3.5=140%2A2%2F3.5%2F2=280%2F7=highlight%2840%29 ALIGN=MIDDLE  ALT=\"140%2F3.5=140%2A2%2F3.5%2F2=280%2F7=highlight%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE ALGEBRA WAY WITH ONE VARIABLE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= speed of the slower second car (in mph, miles per hour)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B16 ALIGN=MIDDLE  ALT=\"x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" >= speed of the faster first car (in mph, miles per hour)<BR>\n" );
document.write( "The time it took for each car to complete the test can be calculated as the distance covered, divided by the speed,<BR>\n" );
document.write( "Since it was the same for both cars,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196%2F%28x%2B16%29=140%2Fx ALIGN=MIDDLE  ALT=\"196%2F%28x%2B16%29=140%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x=140%28x%2B16%29 ALIGN=MIDDLE  ALT=\"196x=140%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Possible explanations for that step:<BR>\n" );
document.write( "They are equivalent equations, and if we have to \"show our work\" more clearly<BR>\n" );
document.write( "we use whatever explanation/justification the teacher thinks is needed.<BR>\n" );
document.write( "If we studied proportions we say that the equation above is a proportion,<BR>\n" );
document.write( "and if we were told that to solve proportions we<BR>\n" );
document.write( "\"equate the cross products\" we just \"cross-multiply\" and write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x=140%28x%2B16%29 ALIGN=MIDDLE  ALT=\"196x=140%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Otherwise we say that we get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x=140%28x%2B16%29 ALIGN=MIDDLE  ALT=\"196x=140%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by \"eliminating denominators\",<BR>\n" );
document.write( "and we do that by multiplying both sides of the equal sign times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B16%29 ALIGN=MIDDLE  ALT=\"%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in two steps, or multiply times both together in one step, multiplying times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B16%29 ALIGN=MIDDLE  ALT=\"x%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x=140%28x%2B16%29 ALIGN=MIDDLE  ALT=\"196x=140%28x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x=140x%2B2240 ALIGN=MIDDLE  ALT=\"196x=140x%2B2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196x-140x=2240 ALIGN=MIDDLE  ALT=\"196x-140x=2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56x=2240 ALIGN=MIDDLE  ALT=\"56x=2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xS=2240%2F56 ALIGN=MIDDLE  ALT=\"xS=2240%2F56\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=40%29 ALIGN=MIDDLE  ALT=\"highlight%28x=40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B16=40%2B16%29%29%29+--%3E+%7B%7B%7Bhighlight%28x%2B16=56%29 ALIGN=MIDDLE  ALT=\"x%2B16=40%2B16%29%29%29+--%3E+%7B%7B%7Bhighlight%28x%2B16=56%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "ONE ALGEBRA WAY WITH TWO VARIABLES:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= speed of the slower second car (in mph, miles per hour)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B16 ALIGN=MIDDLE  ALT=\"x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" >= speed of the faster first car (in mph, miles per hour)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > time that each car took to complete the test drive (in hours).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2At=140 ALIGN=MIDDLE  ALT=\"x%2At=140\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the distance covered by the slower second car (in miles)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B16%29%2At=196 ALIGN=MIDDLE  ALT=\"%28x%2B16%29%2At=196\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2At%2B16t=196 ALIGN=MIDDLE  ALT=\"x%2At%2B16t=196\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the distance covered by the faster first car (in miles)<BR>\n" );
document.write( "We can solve the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2At=140%2Cx%2At%2B16t=196%29 ALIGN=MIDDLE  ALT=\"system%28x%2At=140%2Cx%2At%2B16t=196%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "by just plugging into the second equation the value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2At ALIGN=MIDDLE  ALT=\"x%2At\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the first equation to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140%2B16t=196 ALIGN=MIDDLE  ALT=\"140%2B16t=196\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t=196-140 ALIGN=MIDDLE  ALT=\"16t=196-140\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t=56 ALIGN=MIDDLE  ALT=\"16t=56\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=56%2F16 ALIGN=MIDDLE  ALT=\"t=56%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3.5 ALIGN=MIDDLE  ALT=\"t=3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we plug that value into the first equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2At=140 ALIGN=MIDDLE  ALT=\"x%2At=140\"   DISABLED_style= \"max-width:90%; height:auto;\" > , to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A3.5=196 ALIGN=MIDDLE  ALT=\"x%2A3.5=196\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=196%2F3.5 ALIGN=MIDDLE  ALT=\"x=196%2F3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=40%29 ALIGN=MIDDLE  ALT=\"highlight%28x=40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B16=40%2B16%29%29%29+--%3E+%7B%7B%7Bhighlight%28x%2B16=56%29 ALIGN=MIDDLE  ALT=\"x%2B16=40%2B16%29%29%29+--%3E+%7B%7B%7Bhighlight%28x%2B16=56%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER ALGEBRA WAY WITH TWO VARIABLES:<BR>\n" );
document.write( "We need names for the variables, and when using two variables we should be allowed to use any name we want. I'll use names that make it easier to remember what we are talking about. Feel free to choose any name you like (as long as you don't think the teacher will object).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > =  speed of First (or Faster) car<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" > =  speed of Second (or Slower) car<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=S%2B16 ALIGN=MIDDLE  ALT=\"F=S%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and we can figure out that, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=time=distance%2Fspeed ALIGN=MIDDLE  ALT=\"time=distance%2Fspeed\"   DISABLED_style= \"max-width:90%; height:auto;\" > was the same for both cars,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196%2FF=140%2FS ALIGN=MIDDLE  ALT=\"196%2FF=140%2FS\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196S=140F ALIGN=MIDDLE  ALT=\"196S=140F\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Then we can solve the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28F=S%2B16%2C196S=140F%29 ALIGN=MIDDLE  ALT=\"system%28F=S%2B16%2C196S=140F%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the second equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196S=140%28S%2B16%29 ALIGN=MIDDLE  ALT=\"196S=140%28S%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196S=140S%2B140%2A16 ALIGN=MIDDLE  ALT=\"196S=140S%2B140%2A16\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196S=140S%2B2240 ALIGN=MIDDLE  ALT=\"196S=140S%2B2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196S-140S=2240 ALIGN=MIDDLE  ALT=\"196S-140S=2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56S=2240 ALIGN=MIDDLE  ALT=\"56S=2240\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=2240%2F56 ALIGN=MIDDLE  ALT=\"S=2240%2F56\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28S=40%29 ALIGN=MIDDLE  ALT=\"highlight%28S=40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );