you have five bales of hay. for some reason,instead of being weighed individually, they were weighed in all possible combinations of two:bales 1 and 2, bales 1 and 3,bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4, and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90, and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know. Try to find an easier and more efficient way to find the weights.
To solve this problem efficiently, let's break it down step by step:
Step 1: Determine the number of bales.
Since all possible combinations of two bales were weighed, we can determine the number of bales by finding the maximum number in the list of weights and adding 1. In this case, the maximum weight is 91, so there are 92 bales.
Step 2: Calculate the total weight.
Add up all the weights given: 80 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 90 + 91 = 856.
Step 3: Calculate the sum of all possible combinations.
Since each weight in the list consists of the sum of two bales, we can calculate the sum of all the bales by dividing the total weight by 2. In this case, 856 / 2 = 428.
Step 4: Determine the weight of each bale.
Now we need to determine the weight of each individual bale. To do this, we need to find the sum of the weight of all bales except for a specific bale. We can subtract each possible weight combination from the sum calculated in Step 3.
For example, to find the weight of Bale 1:
Sum of all weights without Bale 1 = 428 - (82 + 83 + 84 + 85 + 86) = 108.
By repeating this process for each bale, we can find the weight of each individual bale:
Bale 1: 108 kg
Bale 2: 110 kg
Bale 3: 112 kg
Bale 4: 114 kg
Bale 5: 116 kg
Therefore, the weights of the five bales are 108 kg, 110 kg, 112 kg, 114 kg, and 116 kg.
Regarding whether there is more than one possible set of weights, in this case, there is only one possible set of weights since the given weights correspond to a specific pattern. However, if there were multiple sets of weights that satisfied the given conditions, we would need additional information or constraints to determine the specific weights.
To find out how much each bale weighs, we can use a systematic approach. Let's break down the problem step by step:
Step 1: Organize the given weights in pairs according to the bales they represent:
Pairs: (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)
Weights: 80, 82, 83, 84, 85, 86, 87, 88, 90, 91
Step 2: Determine the weight increment between each pair of bales:
- Notice that for each pair, one of the weights corresponds to the sum of the bales' individual weights, while the other weight includes their common weight.
- We can find the common weight by finding the difference between the two weights of each pair.
Determining the weight increments:
(1, 2): 82 - 80 = 2
(1, 3): 83 - 80 = 3
(1, 4): 84 - 80 = 4
(1, 5): 85 - 80 = 5
(2, 3): 85 - 82 = 3
(2, 4): 86 - 82 = 4
(2, 5): 87 - 82 = 5
(3, 4): 88 - 83 = 5
(3, 5): 90 - 83 = 7
(4, 5): 91 - 84 = 7
Step 3: Identify the increments repeated more than once:
- From the increments obtained, we notice that 3 and 5 occur twice each.
Step 4: Determine the common weight using the repeated increments:
- Since 3 and 5 occur twice each, we can deduce that the common weight must be 3 + 5 = 8.
Step 5: Calculate the individual bale weights:
- To find the weight of a specific bale, we need to add half of the common weight to the sum of the weights the bale was involved in.
- Let's calculate the individual bale weights:
Bale 1: Sum of weights (1, 2), (1, 3), (1, 4), (1, 5)
= 2 + 3 + 4 + 5 = 14
Weight of Bale 1 = (14 + 8) / 2 = 11
Bale 2: Sum of weights (1, 2), (2, 3), (2, 4), (2, 5)
= 2 + 3 + 4 + 5 = 14
Weight of Bale 2 = (14 + 8) / 2 = 11
Bale 3: Sum of weights (1, 3), (2, 3), (3, 4), (3, 5)
= 3 + 3 + 5 + 7 = 18
Weight of Bale 3 = (18 + 8) / 2 = 13
Bale 4: Sum of weights (1, 4), (2, 4), (3, 4), (4, 5)
= 4 + 4 + 5 + 7 = 20
Weight of Bale 4 = (20 + 8) / 2 = 14
Bale 5: Sum of weights (1, 5), (2, 5), (3, 5), (4, 5)
= 5 + 5 + 7 + 7 = 24
Weight of Bale 5 = (24 + 8) / 2 = 16
Therefore, the weights of the five bales are as follows:
Bale 1 weighs 11 kilograms.
Bale 2 weighs 11 kilograms.
Bale 3 weighs 13 kilograms.
Bale 4 weighs 14 kilograms.
Bale 5 weighs 16 kilograms.
By following this systematic approach, there is only one possible set of weights for the bales. The uniqueness of the solution is evident in Step 5, where the equation to obtain the weight of each bale clearly depends on the given weights and their repetition pattern, leaving no room for ambiguity.