document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=485024>Question 485024</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the reciprocals of two even integer is 9/40. What are the integers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331822 by </B><A HREF=/tutors/aboutme.mpl?userid=chessace><B>chessace(471)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=chessace><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=chessace><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=331822\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a fun problem!<BR>\n" );
document.write( "1/a + 1/b =<BR>\n" );
document.write( "b/ab + a/ab =<BR>\n" );
document.write( "(a+b)/ab = 9/40<BR>\n" );
document.write( "So there is a constant c, not necessarily 1 such that<BR>\n" );
document.write( "a+b = 9c and ab = 40c<BR>\n" );
document.write( "b=9c-a<BR>\n" );
document.write( "a(9c-a)= 40c<BR>\n" );
document.write( "9ac - a^2 - 40c = 0<BR>\n" );
document.write( "a^2 - 9ca + 40c = 0<BR>\n" );
document.write( "----<BR>\n" );
document.write( "At this point a very interesting thing happened.<BR>\n" );
document.write( "Because I was in a rush and have been out of HS for a LONG time, I wrote<BR>\n" );
document.write( "\"This is no good if c>0\"<BR>\n" );
document.write( "Later I went back and used the Quadratic Formula to find:<BR>\n" );
document.write( "c=2 a=8 b=10<BR>\n" );
document.write( "I then recalled enjoying this same problem in HS.<BR>\n" );
document.write( "It's possible that at that time the problem was stated \"... 2 consecutive even intergers...\" but let's assume not.<BR>\n" );
document.write( "I'm now a much better mathematian than I was in HS.<BR>\n" );
document.write( "In particular, I don't think it would have occured to me to look for a negative c after success with c=2.<BR>\n" );
document.write( "This is somewhat of a warning in that if you turn this into a HS Math teacher, it may raise 2 red flags that it was not done by you.<BR>\n" );
document.write( "Back to the original solution.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Let d = -c<BR>\n" );
document.write( "a^2 + 9da - 40d = 0<BR>\n" );
document.write( "To factor this, we need 2 numbers whose difference is 9d and product is 40d.<BR>\n" );
document.write( "There might be a direct way to do this, but good old trial and error (d = 1 2 3 4)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Again, had I had time to find the more direct way, I would have skipped all odd d.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "which finds d=4, numbers 4 and 40: product 160, difference 36.<BR>\n" );
document.write( "a^2 + 36a - 160 = 0<BR>\n" );
document.write( "(a + 40)(a - 4) = 0<BR>\n" );
document.write( "a = -40, 4.<BR>\n" );
document.write( "b = -36 - a = -36 - (-40 or 4) = 4 or -40; of course the same set (excellent double check).<BR>\n" );
document.write( "The 2 numbers are 4 and -40<BR>\n" );
document.write( "Triple check: 1/4 - 1/40 = 10/40 - 1/40 = 9/40<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );