document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162693>Question 1162693</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On a 12-hour clock, at what precise times (in fractions of a minute or second) would the hands be at right angles (90 degrees) between 10 and 11 o'clock? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786528 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=786528\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There will be two times.  A few minutes after 10 o'clock, the minute hand will be 90 degrees \"ahead of\" the hour hand; and a few minutes after 10:30 the minute hand will be 90 degrees \"behind\" the hour hand.<br><BR>\n" );
document.write( "There are many ways to solve this problem.  I will show you the way I find easiest; perhaps other tutors will respond showing different ways.<br><BR>\n" );
document.write( "In every 12-hour period, the angle between the hour hand and minute hand will be any fixed number of degrees 11 times.  That is because the minute hand makes 12 revolutions in 12 hours while the hour hand makes only one.<br><BR>\n" );
document.write( "So for any fixed measure between the hands of a clock, the times when the hands are at that angle will differ by 12/11 of an hour.<br><BR>\n" );
document.write( "The minute hand is 90 degrees ahead of the hour hand at 9 o'clock.  So the time shortly after 10 o'clock when it will be 90 degrees ahead of the hour hand will be 9 hours plus 12/11 hours, or 10 and 1/11 hours.  In terms of fractions of a minute, that means 60/11 minutes after 10 o'clock.<br><BR>\n" );
document.write( "Similarly, the minute hand is 90 degrees behind the hour hand at 3 o'clock.  So the time between 10 and 11 that it is again 90 degrees behind the hour hand is 3 o'clock, plus 7 time 12/11 hours.  That makes 3 + 84/11 = 3 + 7 7/11 hours = 10 7/11 hours.  In terms of fractions of a minute, that is 420/11 minutes after 10 o'clock.<br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "(1) 10 o'clock plus 60/11 minutes, or 5 5/11 minutes after 10.<BR>\n" );
document.write( "(2) 10 o'clock plus 420/11 minutes, or 38 2/11 minutes after 10.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );