document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174984>Question 1174984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a three-digit number such that the hundreds’ digit minus the tens’ digit is 1, the tens’ digit<BR>\n" );
document.write( "minus the units’ digit is 1, and the sum of the digits is 15. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800493 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=800493\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "h = hundreds digit<BR>\n" );
document.write( "t = tens digit<BR>\n" );
document.write( "u = units digit\r<BR>\n" );
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document.write( "A three digit number like 789 means that h = 7, t = 8, u = 9<BR>\n" );
document.write( "We can also construct that number like so<BR>\n" );
document.write( "789 = 700+80+9<BR>\n" );
document.write( "789 = 7*100+8*10+9*1\r<BR>\n" );
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document.write( "Going back to the general format, the three digit number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=htu ALIGN=MIDDLE  ALT=\"htu\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100h%2B10t%2Bu ALIGN=MIDDLE  ALT=\"100h%2B10t%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "As another example, let's say <BR>\n" );
document.write( "h = 5<BR>\n" );
document.write( "t = 4<BR>\n" );
document.write( "u = 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=htu+=+100h%2B10t%2Bu ALIGN=MIDDLE  ALT=\"htu+=+100h%2B10t%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=542+=+100%2A5%2B10%2A4%2B2 ALIGN=MIDDLE  ALT=\"542+=+100%2A5%2B10%2A4%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=542+=+500%2B40%2B2 ALIGN=MIDDLE  ALT=\"542+=+500%2B40%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=542+=+542 ALIGN=MIDDLE  ALT=\"542+=+542\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In this very narrow context, the notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=htu ALIGN=MIDDLE  ALT=\"htu\"   DISABLED_style= \"max-width:90%; height:auto;\" > does <u>not</u> mean \"h times t times u\", even though with many algebra problems, we're conditioned to see two letters together and think \"that's implied multiplication\". \r<BR>\n" );
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document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=htu ALIGN=MIDDLE  ALT=\"htu\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I'm simply using three blank placeholders as the three digits.\r<BR>\n" );
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document.write( "Anyways, we're told that \"hundreds’ digit minus the tens’ digit is 1\" which means we can form the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h-t+=+1 ALIGN=MIDDLE  ALT=\"h-t+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solving for h gets us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+1%2Bt ALIGN=MIDDLE  ALT=\"h+=+1%2Bt\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We're also told that \"the tens’ digit minus the units’ digit is 1\" which forms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-u=1 ALIGN=MIDDLE  ALT=\"t-u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that solves to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+1%2Bu ALIGN=MIDDLE  ALT=\"t+=+1%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The two equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+1%2Bt ALIGN=MIDDLE  ALT=\"h+=+1%2Bt\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+1%2Bu ALIGN=MIDDLE  ALT=\"t+=+1%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be combined like so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+1%2Bt ALIGN=MIDDLE  ALT=\"h+=+1%2Bt\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+1%2B%281%2Bu%29 ALIGN=MIDDLE  ALT=\"h+=+1%2B%281%2Bu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace t with 1+u<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+2%2Bu ALIGN=MIDDLE  ALT=\"h+=+2%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "So whatever the hundreds digit (h) is, we add on 2 to get the units digit (u).\r<BR>\n" );
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document.write( "The last piece of useful information is that \"the sum of the digits is 15\". So the three digits add to 15.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2Bt%2Bu+=+15 ALIGN=MIDDLE  ALT=\"h%2Bt%2Bu+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2B%281%2Bu%29%2Bu+=+15 ALIGN=MIDDLE  ALT=\"h%2B%281%2Bu%29%2Bu+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+1%2Bu ALIGN=MIDDLE  ALT=\"t+=+1%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2B1%2B2u+=+15 ALIGN=MIDDLE  ALT=\"h%2B1%2B2u+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bu%2B1%2B2u+=+15 ALIGN=MIDDLE  ALT=\"2%2Bu%2B1%2B2u+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+2%2Bu ALIGN=MIDDLE  ALT=\"h+=+2%2Bu\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's solve for the variable 'u'.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%2B3+=+15 ALIGN=MIDDLE  ALT=\"3u%2B3+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%2B3-3+=+15-3 ALIGN=MIDDLE  ALT=\"3u%2B3-3+=+15-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u+=+12 ALIGN=MIDDLE  ALT=\"3u+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%2F3+=+12%2F3 ALIGN=MIDDLE  ALT=\"3u%2F3+=+12%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+4 ALIGN=MIDDLE  ALT=\"u+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is the units digit\r<BR>\n" );
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document.write( "Once we know that, we can find the values of t and h<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+1%2Bu+=+1%2B4+=+5 ALIGN=MIDDLE  ALT=\"t+=+1%2Bu+=+1%2B4+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+2%2Bu+=+2%2B4+=+6 ALIGN=MIDDLE  ALT=\"h+=+2%2Bu+=+2%2B4+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We have <BR>\n" );
document.write( "h = 6<BR>\n" );
document.write( "t = 5<BR>\n" );
document.write( "u = 4\r<BR>\n" );
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document.write( "To check, we see that<BR>\n" );
document.write( "h+t+u = 6+5+4 = 11+4 = 15<BR>\n" );
document.write( "so that confirms the third equation and statement.\r<BR>\n" );
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document.write( "Also, we can see that the difference from the hundreds and tens digit (6 and 5) is 1 unit, so is the difference from the tens to the units digit. So the answer is fully confirmed.\r<BR>\n" );
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document.write( "Side note: you could use a trial and error process to guess three digit numbers, and see if you can land on the answer. But I think the algebraic method is most efficient.\r<BR>\n" );
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document.write( "------------------------------------------------\r<BR>\n" );
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document.write( "Answer: 654<BR>\n" );
document.write( "</font> </SPAN>\n" );
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