document.write( "Question 1182878: Five friends Adam, Bipin, Chen, Dana and Erica all take a maths test. Their
\n" ); document.write( "teacher decides not to tell them their individual marks but gives them the
\n" ); document.write( "following information:
\n" ); document.write( "-Adam, Bipin and Chen together scored 200 marks.
\n" ); document.write( "-Bipin, Chen and Dana together scored 215 marks.
\n" ); document.write( "-Chen, Dana and Erica together scored 224 marks.
\n" ); document.write( "-The two girls (Dana and Erica) scored more than Chen.
\n" ); document.write( "-The five of them together scored 350 marks and each of them scored a whole
\n" ); document.write( "number of marks.
\n" ); document.write( "What were their individual scores?
\n" ); document.write( "I can only solve Chen, which is 74. Then, I don't know how to solve. \n" ); document.write( "
Algebra.Com's Answer #812997 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
-Adam, Bipin and Chen together scored 200 marks.
A+B+C = 200
-Bipin, Chen and Dana together scored 215 marks.
B+C+D = 215
-Chen, Dana and Erica together scored 224 marks.
C+D+E = 224
-The two girls (Dana and Erica) scored more than Chen.
D > C and E > C
-The five of them together scored 350 marks
 A+B+C+D+E=350\r\n" );
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document.write( "So we have this system of 4 equations in 5 unknowns,\r\n" );
document.write( "which is \"under-determined\":\r\n" );
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document.write( "A+B+C     = 200\r\n" );
document.write( "  B+C+D   = 215\r\n" );
document.write( "    C+D+E = 224\r\n" );
document.write( "A+B+C+D+E = 350\r\n" );
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document.write( "We can reduce it to a 3x3 system by letting \r\n" );
document.write( "A+B=F and D+E=G in the 1st, 3rd, and 4th equations. \r\n" );
document.write( "Then we have this 3x3 system:\r\n" );
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document.write( "F+C   = 200\r\n" );
document.write( "  C+G = 224\r\n" );
document.write( "F+C+G = 350\r\n" );
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document.write( "Subtracting the 1st eq. from the 3rd gives G = 150 \r\n" );
document.write( "Subtracting the 2nd eq. from the 3rd gives F = 126\r\n" );
document.write( "Substitution of 126 for F in the 1st gives C = 74\r\n" );
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document.write( "So A+B = 126, D+E = 150, and C = 74.\r\n" );
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document.write( "Since we know that D and E are both greater than C=74,\r\n" );
document.write( "let m be how much greater D is than C=74, and\r\n" );
document.write( "let n be how much greater E is than C=74. then\r\n" );
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document.write( "D = 74+m\r\n" );
document.write( "E = 74+n, where m and n are positive integers.\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "             D+E = 150\r\n" );
document.write( "   (74+m)+(74+n) = 150\r\n" );
document.write( "       74+m+74+n = 150\r\n" );
document.write( "         148+m+n = 150\r\n" );
document.write( "             m+n = 2\r\n" );
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document.write( "The only positive integer solution to that is\r\n" );
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document.write( "         m = 1 and n = 1.\r\n" );
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document.write( " So D = 74+m = 74+1 = 75\r\n" );
document.write( "And E = 74+n = 74+1 = 75\r\n" );
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document.write( "We find B from B+C+D = 215\r\n" );
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document.write( "B+74+75 = 215\r\n" );
document.write( "  B+149 = 215\r\n" );
document.write( "      B = 66\r\n" );
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document.write( "We find A from A+B = 126\r\n" );
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document.write( "A+66 = 126\r\n" );
document.write( "   A = 60\r\n" );
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document.write( "So (A,B,C,D,E) = (60,66,74,75,75)\r\n" );
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document.write( "Edwin
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