document.write( "Question 54023This question is from textbook Algebra Structure and Method
\n" ); document.write( ": Section title is Puzzle Problems.\r
\n" ); document.write( "\n" ); document.write( "A number between 300 and 400 is forty times the sum of its digits. The tens digit is 6 more than the unit digit. find the number. Found the answer as 360, but can't figure out all of the steps \n" ); document.write( "
Algebra.Com's Answer #36282 by AnlytcPhil(1806) $\"\"$ $\"About$
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A number between 300 and 400 is forty \r\n" );
document.write( "times the sum of its digits. The tens \r\n" );
document.write( "digit is 6 more than the unit digit.\r\n" );
document.write( "find the number. Found the answer as 360,\r\n" );
document.write( "but can't figure out all of the steps\r\n" );
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document.write( "Yes a little trial and error gives the\r\n" );
document.write( "answer.  But we don't want to use any\r\n" );
document.write( "trial and error.  I also, for personal\r\n" );
document.write( "reasons. do not want to use anything given\r\n" );
document.write( "that is not necessary, such as that it is\r\n" );
document.write( "between 300 and 400, which the other tutor \r\n" );
document.write( "used immediately to show that the hundreds\r\n" );
document.write( "digit is 3.  So I won't even assume that.\r\n" );
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document.write( "It's not easy, though.  Here goes:\r\n" );
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document.write( "Let h = hundredths digit\r\n" );
document.write( "Let t = tens digit\r\n" );
document.write( "Let u = units (ones) digit\r\n" );
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document.write( "The first sentence tells us\r\n" );
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document.write( "100h + 10t + u = 40(h + t + u)\r\n" );
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document.write( "100h + 10t + u = 40h + 40t + 40u\r\n" );
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document.write( "60h = 30t + 39u\r\n" );
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document.write( "which can be divided thru by 3\r\n" );
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document.write( "20h = 10t + 13u\r\n" );
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document.write( "The second sentence tells us\r\n" );
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document.write( "t = u + 6\r\n" );
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document.write( "Substituting in 20h = 10t + 13u\r\n" );
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document.write( "20h = 10(u + 6) + 13u\r\n" );
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document.write( "20h = 10u + 60 + 13u\r\n" );
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document.write( "20h = 23u + 60\r\n" );
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document.write( "Write 23u in terms of its nearest \r\n" );
document.write( "multiple of 20, which is 20u + 3u\r\n" );
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document.write( "20h = 20u + 3u + 60\r\n" );
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document.write( "Divide thru by 20\r\n" );
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document.write( "h = u + 3u/20 + 3\r\n" );
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document.write( "h - u - 3 = 3u/20\r\n" );
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document.write( "Since the left side is an integer, \r\n" );
document.write( "so is the right side\r\n" );
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document.write( "Suppose that integer is A, then\r\n" );
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document.write( "3u/20 = A\r\n" );
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document.write( "3u = 20A\r\n" );
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document.write( "Write 20A in terms of its nearest multiple \r\n" );
document.write( "of 3, which is 21A - A\r\n" );
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document.write( "3u = 21A - A\r\n" );
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document.write( "Divide thru by 3\r\n" );
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document.write( "u = 7A - A/3\r\n" );
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document.write( "A/3 = 7A - u\r\n" );
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document.write( "The right side is an integer, so the \r\n" );
document.write( "left side must be also.\r\n" );
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document.write( "Suppose that integer is B\r\n" );
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document.write( "A/3 = B\r\n" );
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document.write( "A = 3B\r\n" );
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document.write( "Substitute that in 3u = 20A\r\n" );
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document.write( "3u = 20(3B)\r\n" );
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document.write( "3u = 60B\r\n" );
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document.write( "u = 20B\r\n" );
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document.write( "Now since 0 < u < 9\r\n" );
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document.write( "0 < 20B < 9\r\n" );
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document.write( "Divide thru by 20\r\n" );
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document.write( "0 < B < 9/20\r\n" );
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document.write( "0 < B < 0.45\r\n" );
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document.write( "Since B is an integer, \r\n" );
document.write( "it can only be 0\r\n" );
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document.write( "B = 0\r\n" );
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document.write( "substitute in u = 20B\r\n" );
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document.write( "u = 20(0)\r\n" );
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document.write( "u = 0\r\n" );
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document.write( "Substitute in  h = u + 3u/20 + 3\r\n" );
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document.write( "h = 0 + 3(0)/20 + 3\r\n" );
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document.write( "h = 3\r\n" );
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document.write( "Sunstitute u = 0 also in t = u + 6\r\n" );
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document.write( "t = 0 + 6\r\n" );
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document.write( "t = 6\r\n" );
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document.write( "So h = 3, t = 6, and u = 0\r\n" );
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document.write( "So the number is 360.\r\n" );
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document.write( "Notice that we did not need to be given\r\n" );
document.write( "that the number was between 300 and 400. \r\n" );
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document.write( "Edwin
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