document.write( "Question 697898: when joe was half as old as kate. his cousin jean was 38 when kate was half as old as jean. then joe himself was 17 their ages total 113 so what must those three ages be \n" ); document.write( "
Algebra.Com's Answer #430358 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You will notice that I have changed Jean's name to Marie, since Joe and Jean\r\n" );
document.write( "start with the same letter.  It makes age problems easier to think through\r\n" );
document.write( "if you can use people's initials to represent their ages.\r\n" );
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\n" ); document.write( "When Joe was half as old as Kate, his cousin Marie was 38. When Kate was half as old as Marie, then Joe himself was 17. Their ages total 113, so what must those three ages be?
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document.write( "In addition to their ages, we must introduce additional\r\n" );
document.write( "variables for the two 'numbers of years ago' implied in the\r\n" );
document.write( "problem.  We will let them be X years ago and Y years ago,\r\n" );
document.write( "respectively.\r\n" );
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document.write( "We begin by interpreting these words:\r\n" );
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\n" ); document.write( ">>..When Joe was half as old as Kate..<<
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document.write( "X YEARS AGO WAS when Joe was half as old as Kate WAS X YEARS AGO:\r\n" );
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document.write( "Joe was J-X then and Kate was K-X then, so\r\n" );
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document.write( "J-X = \"1%2F2\"(K-X)\r\n" );
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document.write( "Next we interpret the words:\r\n" );
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\n" ); document.write( ">>..His cousin Marie was 38..<<
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\r\n" );
document.write( "We are still talking about X years ago:\r\n" );
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document.write( "X YEARS AGO WAS WHEN his cousin Marie was 38.\r\n" );
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document.write( "Marie was M-X years then, so\r\n" );
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document.write( "M-X = 38  \r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>..When Kate was half as old as Marie,..<<
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\r\n" );
document.write( "Now we're talking about Y years ago.\r\n" );
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document.write( "Y YEARS AGO WAS when Kate was half as old as Marie. \r\n" );
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document.write( "Kate was K-Y then and Marie was M-Y \r\n" );
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document.write( "So\r\n" );
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document.write( "K-Y = \"1%2F2\"(M-Y)\r\n" );
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document.write( "Next we look at this:\r\n" );
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\n" ); document.write( ">>..then Joe himself was 17..<<
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\r\n" );
document.write( "We are still talking about Y years ago.\r\n" );
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document.write( "so Y YEARS AGO, Joe himself was 17.  \r\n" );
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document.write( "Joe was J-Y then so,\r\n" );
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document.write( "J-Y = 17\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>..Their ages total 113,..<<
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document.write( "That's easy!\r\n" );
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document.write( "Their ages NOW total 113   \r\n" );
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document.write( "So\r\n" );
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document.write( "J+K+M = 113\r\n" );
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document.write( "So we have this system of 5 equations in\r\n" );
document.write( "5 unknowns:\r\n" );
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document.write( "J-X = \"1%2F2\"(K-X)\r\n" );
document.write( "M-X = 38\r\n" );
document.write( "K-Y = \"1%2F2\"(M-Y)\r\n" );
document.write( "J-Y = 17\r\n" );
document.write( "J+K+M = 113\r\n" );
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document.write( "Simplify the first equation:\r\n" );
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document.write( "J-X = \"1%2F2\"(K-X)\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "    2J - 2X = K - X\r\n" );
document.write( "2J - K - X = 0\r\n" );
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document.write( "Simplify the third equation\r\n" );
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document.write( "K-Y = \"1%2F2\"(M-Y}}}\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "2K - 2Y = M - Y\r\n" );
document.write( "2K - M - Y = 0\r\n" );
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document.write( "So here are the 5 equations:\r\n" );
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document.write( "2J - K - X = 0\r\n" );
document.write( "M - X = 38\r\n" );
document.write( "2K - M - Y = 0\r\n" );
document.write( "J - Y = 17\r\n" );
document.write( "J + K + M = 113\r\n" );
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document.write( "Now rearrange the terms so that like letters \r\n" );
document.write( "are in the same columns:\r\n" );
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document.write( "(1)  2J -  K     - X     =   0\r\n" );
document.write( "(2)            M - X     =  38\r\n" );
document.write( "(3)       2K - M     - Y =   0\r\n" );
document.write( "(4)   J              - Y =  17\r\n" );
document.write( "(5)   J +  K + M         = 113\r\n" );
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document.write( "Solve (4) for Y \r\n" );
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document.write( "J - 17 = Y\r\n" );
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document.write( "and substitute in (3)\r\n" );
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document.write( "  2K - M - (J - 17) = 0\r\n" );
document.write( "    2K - M - J + 17 = 0\r\n" );
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document.write( "Rearrange as\r\n" );
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document.write( "(6)  -J + 2K - M         = -17\r\n" );
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document.write( "So take out (3) and (4) and put in (6)\r\n" );
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document.write( "(1)  2J -  K     - X  =   0\r\n" );
document.write( "(2)            M - X  =  38\r\n" );
document.write( "(6)  -J + 2K - M      = -17\r\n" );
document.write( "(5)   J +  K + M      = 113\r\n" );
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document.write( "Solve (2) for X\r\n" );
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document.write( "M - 38 = X\r\n" );
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document.write( "and substitute in (1)\r\n" );
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document.write( "2J - K - (M - 38) = 0\r\n" );
document.write( "2J - K - M + 38 = 0\r\n" );
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document.write( "Rearrange as \r\n" );
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document.write( "(7) 2J - K - M = -38\r\n" );
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document.write( "Take out (1) and (2) and put in (7)\r\n" );
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document.write( "(7)  2J -  K - M = -38\r\n" );
document.write( "(6)  -J + 2K - M = -17\r\n" );
document.write( "(5)   J +  K + M = 113\r\n" );
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document.write( "Add equations (6) and (5) and get\r\n" );
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document.write( "          3K     =  96\r\n" );
document.write( "               K = 32, so Kate is 32\r\n" );
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document.write( "Add equations (7) and (5) and get\r\n" );
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document.write( "     3J          =  75\r\n" );
document.write( "               J = 25, so Joe is 25\r\n" );
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document.write( "Substitute J = 25 and K = 32 in (5)\r\n" );
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document.write( "     25 + 32 + M = 113\r\n" );
document.write( "          57 + M = 113\r\n" );
document.write( "               M =  56 so Marie is 56.\r\n" );
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document.write( "That's the answer.  Now to check it, we will \r\n" );
document.write( "need X and Y, which we can get by going\r\n" );
document.write( "back where we solved for them.  \r\n" );
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document.write( " M - 38 = X and  J - 17 = Y\r\n" );
document.write( "56 - 38 = X     25 - 17 = Y \r\n" );
document.write( "     18 = X          8 = Y\r\n" );
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document.write( "So\r\n" );
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document.write( "X = 18 YEARS AGO and  Y = 8 YEARS AGO.\r\n" );
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document.write( "Let's check the words, not the equations:\r\n" );
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document.write( "

\n" ); document.write( "When Joe was half as old as Kate, his cousin Marie was 38.
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document.write( "18 years ago Joe was 25-18=7 years old, Kate was 32-18=14, \r\n" );
document.write( "so it checks that that was when Joe was half Kate's age.\r\n" );
document.write( "It also checks that Marie was 38 because 56-18=38.\r\n" );
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\n" ); document.write( "When Kate was half as old as Marie, then Joe himself was 17.
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document.write( "8 years ago Kate was 32-8=24, and Marie was 56-8=48,\r\n" );
document.write( "so it checks that that was when Kate was half of Marie's age.\r\n" );
document.write( "It also checks that Joe was 17 because 25-8=17.\r\n" );
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\n" ); document.write( "Their ages total 113,
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document.write( "That's easy to check: 25+32+56=113.\r\n" );
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document.write( "Edwin
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