document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167077>Question 1167077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bag initially had blue, red, and purple gumballs in the ration 2 : 3 : 5. Let N be the<BR>\n" );
document.write( "number of red gumballs added to the bag, and 2N be the number of purple gumballs<BR>\n" );
document.write( "added to the bag. How many blue gumballs we need to add , in the bag so that the<BR>\n" );
document.write( "probability of drawing a blue gumball, a red gumball and a purple gumball is now 1/6,<BR>\n" );
document.write( "2/6, and 3/6, respectively?<BR>\n" );
document.write( "A) is equal to N<BR>\n" );
document.write( "B) is equal to 2N<BR>\n" );
document.write( "C) is less than N<BR>\n" );
document.write( "D) is greater than N but less than 2N<BR>\n" );
document.write( "E) cannot be determined<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #791727 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=791727\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A curious problem -- I have to wonder if the information given is not correct.<br><BR>\n" );
document.write( "Defining N as the number of red gumballs added to the bag suggests that N should be a positive integer; but it turns out to be negative....<br><BR>\n" );
document.write( "Given the original ratio 2:3:5...<br><BR>\n" );
document.write( "let 2x = number of blue originally<BR>\n" );
document.write( "let 3x = number of red originally<BR>\n" );
document.write( "let 5x = number of purple originally<br><BR>\n" );
document.write( "let A be the number of blue to be added.  Then after adding more gumballs,<br><BR>\n" );
document.write( "2x+A = number of blue<BR>\n" );
document.write( "3x+N = number of red<BR>\n" );
document.write( "5x+2N = number of purple<br><BR>\n" );
document.write( "After more gumballs are added, the probability of drawing a blue is 1/6, the probability of drawing a red is 2/6, and the probability of drawing a purple is 3/6.  That means the number of red is 2 times the number of blue, and the number of purple is 3 times the number of blue.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2BN+=+2%282x%2BA%29 ALIGN=MIDDLE  ALT=\"3x%2BN+=+2%282x%2BA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2BN+=+4x%2B2A ALIGN=MIDDLE  ALT=\"3x%2BN+=+4x%2B2A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+N-2A ALIGN=MIDDLE  ALT=\"x+=+N-2A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B2N+=+3%282x%2BA%29 ALIGN=MIDDLE  ALT=\"5x%2B2N+=+3%282x%2BA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B2N+=+6x%2B3A ALIGN=MIDDLE  ALT=\"5x%2B2N+=+6x%2B3A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2N-3A ALIGN=MIDDLE  ALT=\"x+=+2N-3A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So now<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N-2A+=+2N-3A ALIGN=MIDDLE  ALT=\"N-2A+=+2N-3A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+A ALIGN=MIDDLE  ALT=\"N+=+A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the answer to the problem as stated is answer A: the number of blue gumballs to be added is equal to N.<br><BR>\n" );
document.write( "But now let's put some actual numbers in the problem.<br><BR>\n" );
document.write( "The number of blue gumballs is now 2x+N, and the number of red gumballs is 3x+N.<br><BR>\n" );
document.write( "And since the probability of drawing a red is 2/6 while the probability of drawing a blue is 1/6, the number of red gumballs is twice the number of blue gumballs:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2BN+=+2%282x%2BN%29 ALIGN=MIDDLE  ALT=\"3x%2BN+=+2%282x%2BN%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2BN+=+4x%2B2N ALIGN=MIDDLE  ALT=\"3x%2BN+=+4x%2B2N\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+-x ALIGN=MIDDLE  ALT=\"N+=+-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now what we have is this:<br><BR>\n" );
document.write( "original number of blue gumballs: 2x<BR>\n" );
document.write( "original number of red gumballs: 3x<BR>\n" );
document.write( "original number of purple gumballs: 5x<br><BR>\n" );
document.write( "number of blue gumballs \"added\": -x<BR>\n" );
document.write( "number of red gumballs \"added\": -x<BR>\n" );
document.write( "number of purple gumballs \"added\": -2x<br><BR>\n" );
document.write( "ending number of blue gumballs: 2x-x = x<BR>\n" );
document.write( "ending number of red gumballs: 3x-x = 2x<BR>\n" );
document.write( "ending number of purple gumballs: 5x-2x = 3x<br><BR>\n" );
document.write( "And we see from this that the final condition of the problem is satisfied:<br><BR>\n" );
document.write( "P(blue) = x/6x = 1/6<BR>\n" );
document.write( "P(red) = 2x/6x = 2/6<BR>\n" );
document.write( "P(purple) = 3x/6x = 3/6<br>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );