document.write( "Question 982932: At the local grocery store, lemons are 52 cents each and limes are 21 cents each. How many lemons and limes can you buy for exactly $3.75? \n" ); document.write( "
Algebra.Com's Answer #603752 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Sometimes students post this type of problem in elementary mathematics because\r\n" );
document.write( "they're to do them by guessing and checking, not algebraic methods.  Other\r\n" );
document.write( "times they are posted by college students studying number theory and solving\r\n" );
document.write( "Diophantine equations. I'll assume you are studying number theory and not in\r\n" );
document.write( "lower grades doing problems by \"trial and error\".\r\n" );
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document.write( "So that you'll have to do your own work, I'll solve a problem that is solved\r\n" );
document.write( "exactly the same way but which has different numbers from yours. \r\n" );
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\n" ); document.write( "At the local grocery store, lemons are 57 cents each and limes are 21 cents
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document.write( "Let x = the number of lemons\r\n" );
document.write( "Let y = the number of limes\r\n" );
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document.write( "The Diophantine equation is:\r\n" );
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document.write( "57x + 23y = 605\r\n" );
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document.write( "23 is the least coefficient (in absolute value in case it were negative).\r\n" );
document.write( "Write 57 and 605 in terms of their nearest multiple of 23.\r\n" );
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document.write( "The nearest multiple of 23 to 57 is found by 57/23 = 2.478...\r\n" );
document.write( "round to the nearest whole number 2. So the nearest multiple of 23\r\n" );
document.write( "to 57 is 2×23 = 46.  So write 57 as 46+11.\r\n" );
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document.write( "The nearest multiple of 23 to 605 is found by 605/23 = 26.3...\r\n" );
document.write( "round to the nearest whole number 26. So the nearest multiple of 23\r\n" );
document.write( "to 605 is 26×23 = 598.  So write 605 as 598+7.\r\n" );
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document.write( "57x + 23y = 605 becomes:\r\n" );
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document.write( "(46+11)x + 23y = 598 + 7\r\n" );
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document.write( "46x + 11x + 23y = 598 + 7\r\n" );
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document.write( "Divide through by 23\r\n" );
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document.write( "2x + (11/23)x + y = 26 + 7/23\r\n" );
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document.write( "Isolate fractions on the left:\r\n" );
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document.write( "(11/23)x - 7/23  = -2x - y + 26 \r\n" );
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document.write( "The right side is an integer, and therefore so is the\r\n" );
document.write( "left side.  Let that integer be A. We set both sides = A\r\n" );
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document.write( "(11/23)x - 7/23 = A       -2x - y + 26 = A\r\n" );
document.write( "Clear the fraction:\r\n" );
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document.write( "11x - 7 = 23A\r\n" );
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document.write( "11 is the least coefficient in absolute value\r\n" );
document.write( "Write 7 and 23 in terms of their nearest multiple of 11.\r\n" );
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document.write( "The nearest multiple of 11 to 7 is 11. So write 11 as 7+4.\r\n" );
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document.write( "The nearest multiple of 11 to 23 is 22.  So write 23 as 22+1.\r\n" );
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document.write( "11x - 7 = 23A\r\n" );
document.write( "11x - 7 = (22+1)A\r\n" );
document.write( "11x - 7 = 22A + A\r\n" );
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document.write( "Divide through by 11\r\n" );
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document.write( "x  - 7/11 = 2A + A/11\r\n" );
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document.write( "Isolate fractions on the left:\r\n" );
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document.write( "-7/11 - A/11 = 2A - x\r\n" );
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document.write( "The right side is an integer, and therefore so is the\r\n" );
document.write( "left side.  Let that integer be B. We set both sides = B\r\n" );
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document.write( "-7/11 - A/11 = B        2A - x = B\r\n" );
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document.write( "Clear the fraction:\r\n" );
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document.write( "-7 - A = 11B\r\n" );
document.write( "-A = 11B+7\r\n" );
document.write( " A = -11B-7\r\n" );
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document.write( "Substitute in 2A - x = B\r\n" );
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document.write( "2(-11B-7) - x = B\r\n" );
document.write( "  -22B-14 - x = B\r\n" );
document.write( "           -x = 23B+14\r\n" );
document.write( "            x = -23B-14\r\n" );
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document.write( "Substitute for A and x in\r\n" );
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document.write( "        -2x - y + 26 = A\r\n" );
document.write( "-2(-23B-14) - y + 26 = A\r\n" );
document.write( "   46B + 28 - y + 26 = -11B-7 \r\n" );
document.write( "      46B + 54 - y = -11B-7\r\n" );
document.write( "                -y = -57B-61\r\n" );
document.write( "                 y = 57B+61\r\n" );
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document.write( "      x > 0              y > 0\r\n" );
document.write( "-23B-14 > 0         57B+61 > 0\r\n" );
document.write( "   -23B > 14           57B > -61\r\n" );
document.write( "      B < -14/23         B > -61/57\r\n" );
document.write( "      B < -0.6...        B > -1.07...\r\n" );
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document.write( "           -1.07... < B < -0.6...\r\n" );
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document.write( "  Since B is an integer B=-1\r\n" );
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document.write( " x = -23B-14        y = 57B+61\r\n" );
document.write( " x = -23(-1)-14     y = 57(-1)+61\r\n" );
document.write( " x = 23-14          y = -57+61\r\n" );
document.write( " x = 9              y = 4\r\n" );
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document.write( "So 9 lemons and 4 limes.\r\n" );
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document.write( "Now do yours the same way.\r\n" );
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document.write( "Edwin

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