document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122157>Question 122157</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A two-digit number is 11 times its units digit.  The sum of the digits is 12.  Find the number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #89736 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=89736\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To work this problem, you have to think a little bit about what numbers mean.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You are told that you have a two-digit number. That means the number contains some tens (call<BR>\n" );
document.write( "the number of tens T) and some units (call the number of units U). <BR>\n" );
document.write( ".<BR>\n" );
document.write( "This means that the unknown number is written as TU.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The last part of the problem tells you that the sum of the digits in the number is 12. This <BR>\n" );
document.write( "means that T + U = 12.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since T is the number of tens in the number and U is the number of units in the number, the<BR>\n" );
document.write( "value of the number is T times ten plus U times 1 .... or just 10T + U. [For example, <BR>\n" );
document.write( "the number 23 contains 2 tens plus 3 units.]<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So again, the number consists of 10T + U and the problem tells you that this is equal to<BR>\n" );
document.write( "11 times U. In equation form this is written as:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "10T + U = 11U<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now you have two equations with two unknowns:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "T + U = 12<BR>\n" );
document.write( "10T + U = 11U<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's get the bottom equation in standard form by getting the two terms with the variables<BR>\n" );
document.write( "all on one side of the equation. Do that by subtraction 11U from both sides of the equation.<BR>\n" );
document.write( "When you do that subtraction the bottom equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "10T - 10U = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "and the two equations that we have are now:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "T + U = 12<BR>\n" );
document.write( "10T - 10U = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's plan to solve this by variable elimination. One way we can do that is to multiply<BR>\n" );
document.write( "the entire top equation (both sides and all terms) by 10. This will make the top equation<BR>\n" );
document.write( "become 10T + 10U = 120 and the pair of equations is then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "10T + 10U = 120<BR>\n" );
document.write( "10T - 10U = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If we then add the two equations vertically in columns, notice that the +10U and the -10U <BR>\n" );
document.write( "cancel each other out ... so the terms containing U are gone. The vertical addition results<BR>\n" );
document.write( "in the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "20T = 120<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You can solve this for T by dividing both sides of this equation by 20 to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "T = 120/20 = 6<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This tells us that the number we are looking for contains 6 tens ... so it is in the sixties.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "We can return to the equation T + U = 12 and since we now know that T is 6, this equation<BR>\n" );
document.write( "becomes 6 + U = 12. You can solve this equation for U by subtracting 6 from both sides to<BR>\n" );
document.write( "get rid of the 6 on the left side. When you do that subtraction, the result is U = 6.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So now we know that the number TU is 66.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's check it to see that it satisfies the original problem:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Is the number equal to 11 times its units number? That means, does the number 66 equal 11 times<BR>\n" );
document.write( "the units number of 6. Yes, 11 times 6 is equal to 66.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Does the sum of its digits equal 12? Yes, 6 + 6 does equal 12.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So we have checked the problem and we know for certain that the two-digit number we were asked to<BR>\n" );
document.write( "find is 66.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand the problem and a method that can be used for solving it.<BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );