document.write( "Question 1035895: there are 5 sacks, and they are weighed 2 at a time. Their weights are 11,11.2,11.3,11.4,11.5,11.6,11.7,11.8,12 and 12.1. This is the weight of all the possible outcomes. How heavy are each of the sacks? \n" ); document.write( "

Algebra.Com's Answer #650514 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "there are 5 sacks, and they are weighed 2 at a time.
\n" ); document.write( "Their weights are 11, 11.2, 11.3, 11.4, 11.5, 11.6, 11.7, 11.8, 12 and 12.1.
\n" ); document.write( "This is the weight of all the possible outcomes. How heavy are each of the sacks?
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document.write( "Let us call these sacks by the names \"a\", \"b\", \"c\", \"d\" and \"e\".\r\n" );
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document.write( "And let \"a\" be the lightest, \"b\" is the next by the weight, \"c\" is the next after \"b\", \"d\" is next after \"c\", and \"e\" is \r\n" );
document.write( "the next after \"d\" (and the last).\r\n" );
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document.write( "Since all 10 given outcomes are different numbers, all \"a, \"b\", \"c\", \"d\" and \"e\" have different weights. \r\n" );
document.write( "Had some of the sacks have equal weight, some numbers in the outcome data would be equal, which is not the case.\r\n" );
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document.write( "If you add all outcomes, you will get the sum \r\n" );
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document.write( "11 + 11.2 + 11.3 + 11.4 + 11.5 + 11.6 + 11.7+ 11.8 + 12 + 12.1 = 115.6.\r\n" );
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document.write( "It is the sum of all pairs (a+b), (a+c), (a+d), . . . , (d+e) taken 2 at a time. By grouping and regrouping,\r\n" );
document.write( "you can easily understand that it is nothing else as 4*(a+b+c+d+e), i.e four times the sum of (a+b+c+f+e).\r\n" );
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document.write( "It implies that the sum\r\n" );
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document.write( "a + b + c + d + e =  = 28.9.   (1)\r\n" );
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document.write( "Now, it is clear that the very first outcome 11 (the smallest outcome) is the sum of \"a and \"b\", the two lightest sacks. \r\n" );
document.write( "It can not be nothing else. So, we have \r\n" );
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document.write( "a + b = 11.      (2)\r\n" );
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document.write( "Similarly, it is clear that the very next outcome  11.2  is the sum of \"a\" and \"c\". It simply can not be nothing else. So, we have \r\n" );
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document.write( "a + c = 11.2.    (3)\r\n" );
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document.write( "Next, it is clear that the very last outcome 12.1 (the greatest outcome) is the sum of \"d\" and \"e\", the two heaviest sacks. \r\n" );
document.write( "It can not be nothing else. So, we have \r\n" );
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document.write( "d + e = 12.1.    (4)\r\n" );
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document.write( "Similarly, outcome  12  is the sum of \"c\" and \"e\". It simply can not be nothing else. So, we have \r\n" );
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document.write( "c + e = 12.      (5)\r\n" );
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document.write( "Take the sum of (2) and (3). You will get\r\n" );
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document.write( "a + b + d + e = 11 + 12.1 = 23.1.   (6)\r\n" );
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document.write( "Now distract (6) from (1). You will get\r\n" );
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document.write( "c = 28.9 - 23.1 = 5.8.\r\n" );
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document.write( "So, we just found \"c\".\r\n" );
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document.write( "Now we can easily find all remaining unknowns.\r\n" );
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document.write( "From (3),  a = 11.2 - c = 11.2 - 5.8 = 5.4.\r\n" );
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document.write( "From (2),  b = 11   - a = 11   - 5.4 = 5.6.\r\n" );
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document.write( "From (5),  e = 12   - c = 12   - 5.8 = 6.2.\r\n" );
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document.write( "From (4),  d = 12.1 - e = 12.1 - 6.2 = 5.9.\r\n" );
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document.write( "Answer. The sacks are 5.4, 5.6, 5.8, 5.9 and 6.2, from lightest to heaviest.\r\n" );
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