document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=76661>Question 76661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ed and Dave are going to divide a case prize of $14,000 in a ratio of 2 to 5. How much will Dave receive?\r<BR>\n" );
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document.write( "I came up with 8,000. I am not sure this is correct. Could you verify this for me? Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #54983 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=54983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well, let's see.<BR>\n" );
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document.write( "The total prize is $14,000.  Let's call Dave's share D and Ed's share E.  The sum of the two<BR>\n" );
document.write( "shares has to total the entire prize.  So we can write the equation:<BR>\n" );
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document.write( "D + E = 14,000<BR>\n" );
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document.write( "And by subtracting D from both sides of this equation we can see that Ed's share is:<BR>\n" );
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document.write( "E = 14,000 - D<BR>\n" );
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document.write( "So we can set up the ratio of Ed's share (14,000 - D) to Dave's share (D) and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%2FD+=+%2814000+-+D%29%2FD+ ALIGN=MIDDLE  ALT=\"E%2FD+=+%2814000+-+D%29%2FD+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "but the problem also tells us that this ratio is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5 ALIGN=MIDDLE  ALT=\"2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So we can set up the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814000+-+D%29%2FD+=+2%2F5 ALIGN=MIDDLE  ALT=\"%2814000+-+D%29%2FD+=+2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When you have two ratios equal to each other, a method of solving them is to find the<BR>\n" );
document.write( "cross products and set them equal.  The cross products come from multiplying the numerator of<BR>\n" );
document.write( "one ratio times the denominator of the other ratio.  Then multiply the denominator of <BR>\n" );
document.write( "the first ratio times the numerator of the second ratio.<BR>\n" );
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document.write( "In this problem the first cross product is <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814000+-+D%29%2A+5 ALIGN=MIDDLE  ALT=\"%2814000+-+D%29%2A+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--- numerator of first ratio times denominator of second<BR>\n" );
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document.write( "and this multiplication results in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70000+-+5%2AD ALIGN=MIDDLE  ALT=\"70000+-+5%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "and the second cross product is the denominator of the first ratio (D) times the numerator of<BR>\n" );
document.write( "the second ratio (2) to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2AD ALIGN=MIDDLE  ALT=\"+2%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "[You may be able to see why they are often referred to as cross products.  You are multiplying<BR>\n" );
document.write( "along diagonals drawn through the equal sign between the two ratios.]<BR>\n" );
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document.write( "Anyhow, set the two cross products equal and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70000+-+5%2AD+=+2%2AD ALIGN=MIDDLE  ALT=\"70000+-+5%2AD+=+2%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2AD ALIGN=MIDDLE  ALT=\"5%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to eliminate the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2AD ALIGN=MIDDLE  ALT=\"-5%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.  This<BR>\n" );
document.write( "results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70000+=+7%2AD ALIGN=MIDDLE  ALT=\"70000+=+7%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for Dave's share by dividing both sides by 7 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+10000+=+D ALIGN=MIDDLE  ALT=\"+10000+=+D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So Dave gets $10,000 which means that Ed gets the remaining $4,000 of the prize.<BR>\n" );
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document.write( "Now check to see that the ratio is correct.  Ask yourself, is $4,000 divided by $10,000<BR>\n" );
document.write( "the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5 ALIGN=MIDDLE  ALT=\"2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >?  If you divide both the numerator and the denominator by 2,000 you<BR>\n" );
document.write( "find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4000%2F10000 ALIGN=MIDDLE  ALT=\"4000%2F10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > does, in fact, reduce to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5 ALIGN=MIDDLE  ALT=\"2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Therefore, our answer is<BR>\n" );
document.write( "correct.<BR>\n" );
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document.write( "If you do the same verification with your answer in which Dave got $8000 and Ed got the <BR>\n" );
document.write( "remaining $6000 your ratio is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6000%2F8000 ALIGN=MIDDLE  ALT=\"6000%2F8000\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F8 ALIGN=MIDDLE  ALT=\"6%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that reduces to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5 ALIGN=MIDDLE  ALT=\"2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > as required by the problem.<BR>\n" );
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document.write( "Hope this helps you to see your way through the problem and perhaps introduces you to<BR>\n" );
document.write( "using cross products to solve proportions (two ratios set equal). </SPAN>\n" );
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