document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194539>Question 1194539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "In a 3 digit number the units digit is 3 more than the tens digit and the sum of the digits is 11. If the units and hundreds digits are interchanged the number increased  by 99. <BR>\n" );
document.write( "What is original number. <BR>\n" );
document.write( "Thanks </SPAN>\n" );
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document.write( "Answer: <font color=red>425 is the original number</font>\r<BR>\n" );
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document.write( "See below for the work shown and explanation.\r<BR>\n" );
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document.write( "x = hundreds digit<BR>\n" );
document.write( "y = tens digit<BR>\n" );
document.write( "z = units or ones digit<BR>\n" );
document.write( "These variables represent nonnegative single-digit whole numbers from the set {0,1,2,3,4,5,6,7,8,9}<BR>\n" );
document.write( "Though I should be more careful to point out that x = 0 is not possible since something like 012 = 12 is a two digit number. <BR>\n" );
document.write( "Luckily, zero is allowed for the other digits.\r<BR>\n" );
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document.write( "Any three-digit number is of the form 100x+10y+z<BR>\n" );
document.write( "For example, if we had the following three values: <BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "y = 2<BR>\n" );
document.write( "z = 3<BR>\n" );
document.write( "then,<BR>\n" );
document.write( "100x+10y+z = 100*1+10*2+3 = 123\r<BR>\n" );
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document.write( "We're told that \"the units digit is 3 more than the tens digit\", which means symbolically we would say z = y+3<BR>\n" );
document.write( "Whatever y is, add on 3 to get z.\r<BR>\n" );
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document.write( "We also know that \"the sum of the digits is 11\", so,<BR>\n" );
document.write( "x+y+z = 11<BR>\n" );
document.write( "Let's replace z with y+3 and simplify<BR>\n" );
document.write( "x+y+z = 11<BR>\n" );
document.write( "x+y+y+3 = 11<BR>\n" );
document.write( "x+2y+3 = 11<BR>\n" );
document.write( "x+2y = 11-3<BR>\n" );
document.write( "x+2y = 8<BR>\n" );
document.write( "Then we can solve for x<BR>\n" );
document.write( "x+2y = 8<BR>\n" );
document.write( "x = 8-2y<BR>\n" );
document.write( "which will be useful later in a substitution step.\r<BR>\n" );
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document.write( "Let A = 100x+10y+z be the original three-digit number, and B = 100z+10y+x be the reversed version of the original number (eg: A = 123 and B = 321)<BR>\n" );
document.write( "As you can see, I swapped the hundreds digit and units digit to reverse the number.\r<BR>\n" );
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document.write( "We're told that \"if the units and hundreds digits are interchanged the number increased by 99.\"<BR>\n" );
document.write( "meaning,<BR>\n" );
document.write( "new = old+99<BR>\n" );
document.write( "B = A+99<BR>\n" );
document.write( "100z+10y+x = 100x+10y+z+99\r<BR>\n" );
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document.write( "From here, we will plug in these items<BR>\n" );
document.write( "x = 8-2y<BR>\n" );
document.write( "z = y+3<BR>\n" );
document.write( "so that everything is now in terms of y. This will allow us to solve for y.\r<BR>\n" );
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document.write( "100z+10y+x = 100x+10y+z+99<BR>\n" );
document.write( "100(y+3)+10y+8-2y = 100(8-2y)+10y+y+3+99<BR>\n" );
document.write( "100y+300+10y+8-2y = 800-200y+10y+y+3+99<BR>\n" );
document.write( "108y+308 = 902-189y<BR>\n" );
document.write( "108y+189y = 902-308<BR>\n" );
document.write( "297y = 594<BR>\n" );
document.write( "y = 594/297<BR>\n" );
document.write( "y = 2\r<BR>\n" );
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document.write( "Then use this to find x and z<BR>\n" );
document.write( "x = 8-2y = 8-2*2 = 4<BR>\n" );
document.write( "z = y+3 = 2+3 = 5\r<BR>\n" );
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document.write( "The original number is:<BR>\n" );
document.write( "A = 100x+10y+z<BR>\n" );
document.write( "A = 100*4+10*2+5<BR>\n" );
document.write( "A = 400+20+5<BR>\n" );
document.write( "A = <font color=red>425</font><BR>\n" );
document.write( "The reversed number is<BR>\n" );
document.write( "B = 100z+10y+x<BR>\n" );
document.write( "B = 100*5+10*2+4<BR>\n" );
document.write( "B = 500+20+4<BR>\n" );
document.write( "B = 524\r<BR>\n" );
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document.write( "Notice how B-A = 524-425 = 99<BR>\n" );
document.write( "which helps show that B = A+99 is correct.<BR>\n" );
document.write( "Or you could verify like so:<BR>\n" );
document.write( "B = A+99<BR>\n" );
document.write( "524 = 425+99<BR>\n" );
document.write( "524 = 524\r<BR>\n" );
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document.write( "Also, <BR>\n" );
document.write( "x+y+z = 4+2+5 = 6+5 = 11<BR>\n" );
document.write( "confirms the digits add up to 11\r<BR>\n" );
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document.write( "Lastly, the units digit (5) is exactly three more than the tens digit (2).\r<BR>\n" );
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document.write( "Therefore, we have fully confirmed the <font color=red>answer of 425</font><BR>\n" );
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