Something like this:
\documentclass{article}
\begin{document}
Page \pageref{pg:one} of the exam\label{pg:one}.
There are \pageref{pg:examend} pages in this examination.
\clearpage
Page \pageref{pg:two} of the exam\label{pg:two}.
\clearpage
Page \pageref{pg:three} of the exam\label{pg:three}.
\clearpage
Page \pageref{pg:four} of the exam\label{pg:four}.
\clearpage
Page \pageref{pg:five} of the exam\label{pg:five}.
\clearpage
Page \pageref{pg:six} of the exam\label{pg:six}.
\label{pg:examend}
\clearpage
Page \pageref{pg:seven} contains answers\label{pg:seven}.
\end{document}

Please note that I cannot check these images have uploaded correctly as I am unable to view images on the site at the present time.
EDIT: Since you are using the exam class and want to use \numpages, you could adapt the example from the manual of headers and footers to use this as follows:
\documentclass{exam}
\pagestyle{headandfoot}
\runningheadrule
\firstpageheader{Math 115}{First Exam}{July 4, 1776}
\runningheader{Math 115}
{First Exam, Page \thepage\ of \numpages}
{July 4, 1776}
\firstpagefooter{}{}{}
\runningfooter{}{}{}
\def\numpages{%
\pageref{pg:examend}
}% numpages
\begin{document}
Put your exam questions here.
\clearpage
Some more questions here.
\label{pg:examend}
\clearpage
Put the answers here.
\end{document}
The only oddity is that this will give e.g. "Page 3 of 2" on the final page which has the answers. You could avoid that if you liked by redefining the header command at the end of the exam.
lastpage
or the more advancedpageslts
?pageslts
there’s a section “Alternatives”, that you should read. Some of the packages look promising.