Question 9355: supose 5 bales of hay are weighed 2 at a time in all possible ways. the weights in pounds are 110,112,113,114,114,116,117,118,120,and 121. how much does each bale weigh?
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! First of all,you did two mistakes
1) wrong category of posting
2) the numbers are 110,112,113,114,115,116,117,118,120,and 121
[No two 114s, one should be 115. otherwise no legal soutions]
Consider a pentagon , each vertex can be connected to four of the
other vertices. Hence, the summation of the 10 numbers should be
equal to 4 times the sum of the five bales.
Since the total of these 10 numbers is 1156, the sum of the 5 balse is
1156/4 = 289 [use Excel]
Let the five bales be A,B,C,D and E (in this order, ie A is the lightest,
E is the heaviest)
Clearly ,we know that A + B = 110 & D+E = 121.
So,we get the middle one C = 289 - 110 - 121 = 58,
and the 2nd number 112 in the listing should be A + C,
hence A = 112 - 58 = 54, and so B = 110 - 54 = 56.
whence, 120 should be C + E, and we obtain E = 120 - 58 = 62.
Finally, D = 121 - 62 = 59.
Now we get the weights are 54,56,58, 59 & 62 pounds.
Of course, you can set up a system of 10 linear equations in 5 variables
to solve this problem. But, the way above should be much faster.
Or you try to draw a pentagon to help you to understand.
Also, you have to check correctness of the answer.
That is, these 5 numbers do make the given listing.
Kenny
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