document.write( "Question 25006: in a circle i have 8 numbers, a mystery number then 173, 268, 934, 252, 520, 89, and 311. What is the mystery number? \n" ); document.write( "
Algebra.Com's Answer #13418 by AnlytcPhil(1806)\"\" \"About 
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in a cercle i have 8 numbers, a mystery number then 173, 268, 934, 252,\r\n" );
document.write( "520, 89, and 311. What is the mystery number?\r\n" );
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document.write( "Let the mystery number be N.  Then the 8 numbers are\r\n" );
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document.write( "173, 268, 934, 252, 520, 89, 311, and N\r\n" );
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document.write( "Arrange them in a circle like this, putting N for the\r\n" );
document.write( "missing number \r\n" );
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document.write( "          173\r\n" );
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document.write( "      N        268\r\n" );
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document.write( "   311     o     934\r\n" );
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document.write( "     89         252\r\n" );
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document.write( "          520\r\n" );
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document.write( "We look for a pattern.\r\n" );
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document.write( "Compare the numbers that are directly opposite each\r\n" );
document.write( "other in the circle.\r\n" );
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document.write( "173 is opposite 520\r\n" );
document.write( "268 is opposite 89\r\n" );
document.write( "934 is opposite 311\r\n" );
document.write( "252 is opposite N\r\n" );
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document.write( "I first notice that \r\n" );
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document.write( "(1)  520 is a little bit more than three times 173\r\n" );
document.write( "     since 520/173 = 3.005780347\r\n" );
document.write( "(2)  268 is a little bit more than three times 89  \r\n" );
document.write( "     since 268/69 = 3.011235955\r\n" );
document.write( "(3)  934 is a little bit more than three times 311 \r\n" );
document.write( "     since 934/311 = 3.003215434\r\n" );
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document.write( "So I would guess either:\r\n" );
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document.write( "(A) that 252 is a little bit more than three times N\r\n" );
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document.write( "or\r\n" );
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document.write( "(B) that N is a little bit more than three times 252.\r\n" );
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document.write( "But that doesn't answer my question.\r\n" );
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document.write( "So then I ask \"In each case, just how much more is the\r\n" );
document.write( "larger than 3 times the smaller?\r\n" );
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document.write( "Let's find out:\r\n" );
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document.write( "(1)  173 is opposite 520, which is a little more than\r\n" );
document.write( "     3 times 173. How much more? Well 3×173 is 519, and\r\n" );
document.write( "     520 is 1 more than 519.\r\n" );
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document.write( "(2)  89 is opposite 268, which is a little more than \r\n" );
document.write( "     3 times 89. How much more? Well 3×89 is 267, and \r\n" );
document.write( "     268 is 1 more than 267.\r\n" );
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document.write( "(3)  934 is opposite 311, which is a little more than \r\n" );
document.write( "     3 times 311. How much more? Well 3×311 is 933, and \r\n" );
document.write( "     934 is 1 more than 933.\r\n" );
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document.write( "So we have the pattern.  For any pair of opposite numbers in \r\n" );
document.write( "the circle, the larger is 1 more than three times the smaller.  \r\n" );
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document.write( "So I want to find N, the number opposite 252.\r\n" );
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document.write( "What I need to find out now is whether N is the smaller and 252\r\n" );
document.write( "the larger, or whether 252 is the smaller and N the larger. \r\n" );
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document.write( "If N is the smaller then 252 should be 1 more than 3 times N.\r\n" );
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document.write( "Let's see if that's possible.\r\n" );
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document.write( "252 = 3N + 1\r\n" );
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document.write( "252 = 3N\r\n" );
document.write( "   \r\n" );
document.write( "  N = 251/3 = 83.66666...\r\n" );
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document.write( "No, that's not possible since we would expect N to be a whole \r\n" );
document.write( "number like the other seven numbers, not a decimal or fraction.\r\n" );
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document.write( "So 252 is the smaller and should be 1 more than 3 times 252.\r\n" );
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document.write( "3 times 252 is 756 and 1 more than 756 is 757\r\n" );
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document.write( "Let's see if that's possible.\r\n" );
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document.write( "252 = 3N + 1\r\n" );
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document.write( "252 = 3N\r\n" );
document.write( "   \r\n" );
document.write( "  N = 251/3 = 83.66666...\r\n" );
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document.write( "No, that's not possible since we would expect N to be \r\n" );
document.write( "a whole number like all the others, not a decimal or \r\n" );
document.write( "fraction.       \r\n" );
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document.write( "That only leaves the possibility that N is the larger \r\n" );
document.write( "and is 1 more than three times 252.  Three times 252 \r\n" );
document.write( "is 756.  So N must be 1 more than that,\r\n" );
document.write( "so N = 757.\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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