document.write( "Question 402326: For a certain 3-digit number, interchanging the first and last digits gives a 3-digit number 297 less than the original number. Doubling the first and last digits of the original number gives a 3-digit number and increases the sum of the digits to 15. What is the number? \n" ); document.write( "
Algebra.Com's Answer #284630 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
For a certain 3-digit number, interchanging the first and last digits gives a 3-digit number 297 less than the original number. Doubling the first and last digits of the original number gives a 3-digit number and increases the sum of the digits to 15. What is the number?
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document.write( "Let the number have the digits \"htu\" where \r\n" );
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document.write( "h = its hundreds digit\r\n" );
document.write( "t = its tens digit\r\n" );
document.write( "u = its units (ones) digit\r\n" );
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document.write( "The original number = 100h + 10t + u\r\n" );
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\n" ); document.write( ">>...interchanging the first and last digits...<<
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document.write( "That gives 100u + 10t + h \r\n" );
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\n" ); document.write( ">>...gives a 3-digit number 297 less than the original number...<<
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document.write( "That is 100h + 10t + u - 297, so we set them equal\r\n" );
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document.write( "100u + 10t + h = 100h + 10t + u - 297\r\n" );
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document.write( "277 = 99h - 99u \r\n" );
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document.write( "99h - 99u = 277\r\n" );
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document.write( "That equation can be divided through by 99\r\n" );
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document.write( "  h - u = 3\r\n" );
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\n" ); document.write( ">>...Doubling the first and last digits of the original number...<<
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document.write( "That gives a number with hundreds digit 2h, tens digit still t, and\r\n" );
document.write( "units (ones) digit 2u\r\n" );
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\n" ); document.write( ">>...gives a 3-digit number...<<
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document.write( "That tells us that 2h is NOT zero, but we knew that anyway.\r\n" );
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\n" ); document.write( ">>...and increases the sum of the digits to 15...<<
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document.write( "So 2h + t + 2u = 15\r\n" );
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document.write( "So we have the equations:\r\n" );
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document.write( "      h - u = 3\r\n" );
document.write( "2h + t + 2u = 15\r\n" );
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document.write( "Solve the first equation for h\r\n" );
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document.write( "h = 3 + u\r\n" );
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document.write( "Substitute in the second equation:\r\n" );
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document.write( "2(3 + u) + t + 2u = 15\r\n" );
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document.write( "6 + 2u + t + 2u = 15\r\n" );
document.write( "         4u + t = 9\r\n" );
document.write( "              t = 9 - 4u\r\n" );
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document.write( "   Since \r\n" );
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document.write( "              0 < t < 9\r\n" );
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document.write( "            0 < 9 - 4u < 9\r\n" );
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document.write( "            -9 < -4u < 0\r\n" );
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document.write( "             9 < 4u < 0\r\n" );
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document.write( "            9/4 < u < 0 \r\n" );
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document.write( "           2.25 < u < 0 \r\n" );
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document.write( "And since the units (ones) digit must be 1 or larger, because\r\n" );
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\n" ); document.write( ">>...interchanging the first and last digits gives a 3-digit number...<<
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document.write( "so it cannot be 0, for if it were 0, that interchange would make a 0 \r\n" );
document.write( "for the first digit.\r\n" );
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document.write( "So u is either 1 or 2\r\n" );
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document.write( "If u = 1, then since h = 3 + u\r\n" );
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document.write( "h = 3 + 1 = 4\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "2h + t + 2u = 15\r\n" );
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document.write( "2(4) + t + 2(1) = 15\r\n" );
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document.write( "     8 + t + 2 = 15\r\n" );
document.write( "        t + 10 = 15\r\n" );
document.write( "             t = 5\r\n" );
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document.write( "So in this case the number \"htu\" is 451 \r\n" );
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document.write( "Checking:\r\n" );
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document.write( "interchanging the first and last digits gives 154\r\n" );
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document.write( "and that is a 3-digit number 297 less than the original number,\r\n" );
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document.write( "since 451 - 154 = 297 \r\n" );
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\n" ); document.write( ">>...Doubling the first and last digits of the original number gives a 3-digit number and increases the sum of the digits to 15...<<
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document.write( "Doubling the first and last digit of 451 gives 852, and the sum\r\n" );
document.write( "of its digits is 8+5+2=15\r\n" );
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document.write( "So 451 is a solution.\r\n" );
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document.write( "Now we try \r\n" );
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document.write( "If u = 2, then since h = 3 + u\r\n" );
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document.write( "h = 3 + 2 = 5\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "2h + t + 2u = 15\r\n" );
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document.write( "2(5) + t + 2(2) = 15\r\n" );
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document.write( "    10 + t + 4 = 15\r\n" );
document.write( "        t + 14 = 15\r\n" );
document.write( "             t = 1\r\n" );
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document.write( "So in this case the number \"htu\" is 512 \r\n" );
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document.write( "Checking:\r\n" );
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document.write( "interchanging the first and last digits gives 215\r\n" );
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document.write( "and that is not a 3-digit number 297 less than the original number,\r\n" );
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document.write( "since 215 - 512 = -297 \r\n" );
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document.write( "So we must discard this possibility.  Also the last part\r\n" );
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\n" ); document.write( ">>...Doubling the first and last digits of the original number gives a 3-digit number and increases the sum of the digits to 15...<<
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document.write( "You cannot double the first digit 5, for doubling 5 gives 10, \r\n" );
document.write( "not a single digit.\r\n" );
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document.write( "So there is just one solution, 451.\r\n" );
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document.write( "Edwin
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