document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1096548>Question 1096548</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of a, b and c if: a + 1/(b+1/c)  = 37/16 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #710987 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=710987\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>There are an infinite number of solutions if there are no restrictions on the values of a, b, and c.  So I will assume the problem requires the values to be positive integers.  In that case....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+1%2F%28b%2B1%2Fc%29+=+a+%2B+1%2F%28%28bc%2B1%29%2Fc%29+=+a+%2B+c%2F%28bc%2B1%29+=+37%2F16+=+2+%2B+5%2F16 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since a, b, and c have to be positive integers, there are only two possibilities:<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1 ALIGN=MIDDLE  ALT=\"a+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F%28bc%2B1%29+=+21%2F16 ALIGN=MIDDLE  ALT=\"c%2F%28bc%2B1%29+=+21%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >; or<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2 ALIGN=MIDDLE  ALT=\"a+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F%28bc%2B1%29+=+5%2F16 ALIGN=MIDDLE  ALT=\"c%2F%28bc%2B1%29+=+5%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Looking for a solution for case (1)...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F%28bc%2B1%29+=+21%2F16 ALIGN=MIDDLE  ALT=\"c%2F%28bc%2B1%29+=+21%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16c+=+21bc%2B21 ALIGN=MIDDLE  ALT=\"16c+=+21bc%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16c-21bc+=+21 ALIGN=MIDDLE  ALT=\"16c-21bc+=+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2816-21b%29+=+21 ALIGN=MIDDLE  ALT=\"c%2816-21b%29+=+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+21%2F%2816-21b%29 ALIGN=MIDDLE  ALT=\"c+=+21%2F%2816-21b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Clearly there are no positive integer values for b that produce positive integer values for c.  So there are no solutions for this case.<br><BR>\n" );
document.write( "Looking for a solution for case (2)...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F%28bc%2B1%29+=+5%2F16 ALIGN=MIDDLE  ALT=\"c%2F%28bc%2B1%29+=+5%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16c+=+5bc%2B5 ALIGN=MIDDLE  ALT=\"16c+=+5bc%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16c-5bc+=+5 ALIGN=MIDDLE  ALT=\"16c-5bc+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2816-5b%29+=+5 ALIGN=MIDDLE  ALT=\"c%2816-5b%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+5%2F%2816-5b%29 ALIGN=MIDDLE  ALT=\"c+=+5%2F%2816-5b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There is exactly one positive integer value for b that produces a positive integer value for c: b=3 gives c=5.<br><BR>\n" );
document.write( "So there is a unique solution to the problem in positive integers:<BR>\n" );
document.write( "a=2;<BR>\n" );
document.write( "b=3;<BR>\n" );
document.write( "c=5<br><BR>\n" );
document.write( "CHECK:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+1%2F%283%2B1%2F5%29+=+2+%2B+1%2F%2816%2F5%29+=+2+%2B+5%2F16+=+37%2F16 ALIGN=MIDDLE  ALT=\"2+%2B+1%2F%283%2B1%2F5%29+=+2+%2B+1%2F%2816%2F5%29+=+2+%2B+5%2F16+=+37%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The answer to the problem is then a+b+c = 10. </SPAN>\n" );
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