A shipping clerk wishes to determine the weights of each of the 4 boxes. Each box weighs a different integer amount less than 70 kg. Unfortunately, the only scales available measure weights in excess of 70 kg. The clerk therefore decides to weigh the boxes in pairs so that each box is weighed with every other box. The weights for the 6 pairs of boxes are (in kg) 84, 85, 87, 88, 90, and 91. From this information, the clerk can determine the weight of each box.
what are the weights of the four boxes?
I do not know if there are multiple solution, here's what I've found.
The sums of weighed pairs are all correct, but the individual weights are not integers.
A=81/2
B=87/2
C=89/2
D=93/2
A+B=84
A+C=85
A+D=87
B+C=88
B+D=90
C+D=91
To determine the weight of each box, we can use a system of equations.
Let's assign variables to the weights of the boxes. We can use A, B, C, and D to represent the weights of the four boxes, respectively.
Based on the given information, the clerk weighs each pair of boxes and obtains the following results:
A + B = 84
A + C = 85
A + D = 87
B + C = 88
B + D = 90
C + D = 91
Now, we can solve this system of equations to find the values of A, B, C, and D.
We can subtract the second equation (A + C = 85) from the first equation (A + B = 84):
(A + B) - (A + C) = 84 - 85
B - C = -1 ----(Equation 1)
Similarly, subtracting the second equation (A + C = 85) from the third equation (A + D = 87) gives us:
(A + D) - (A + C) = 87 - 85
D - C = 2 ----(Equation 2)
Next, we subtract the fourth equation (B + C = 88) from the fifth equation (B + D = 90):
(B + D) - (B + C) = 90 - 88
D - C = 2 ----(Equation 3)
Comparing Equation 2 and Equation 3, we can see that D - C is equal to 2 in both equations. This means that D and C have the same difference in each pair, implying that D and C have the same values.
Now, we can substitute D = C into Equation 2:
D - C = 2
C - C = 2
0 = 2
This equation is not possible, which means our assumption of D = C is incorrect.
Therefore, we need to make a new assumption: D > C. Since all weights are less than 70 kg, the maximum sum of two weights is 2 * 69 = 138 kg. So, let's assume D > C, and hence, D = C + 1.
Substituting D = C + 1, we can rewrite Equation 3:
D - C = 2
(C + 1) - C = 2
1 = 2
Once again, this equation is not possible.
This means D does not equal C + 1, so we need to make another assumption: D > C + 1. Let's assume D = C + 2.
Substituting D = C + 2 into Equation 3:
D - C = 2
(C + 2) - C = 2
2 = 2
This equation is true, so our assumption D = C + 2 is valid.
Using this assumption, we can find the weights of the boxes:
From Equation 2, we have D - C = 2. Substituting D = C + 2:
(C + 2) - C = 2
2 = 2
This equation is true, which means our assumption D = C + 2 is valid.
Now, let's substitute D = C + 2 into Equation 1:
B - C = -1
Since D = C + 2 and D > C, we have C + 2 > C, which indicates that B > C. So, let's assume B = C + 1 and substitute it into Equation 1:
(B =) (C + 1) - C = -1
1 = -1
This equation is not possible, which means our initial assumption B = C + 1 is incorrect.
To make a new assumption, let's consider B > C + 1. Assuming B = C + 2 and substituting it into Equation 1:
(B =) (C + 2) - C = -1
2 = -1
This equation is not possible. So, we need to revise our assumption again.
Let's assume B = C + 3 and substitute it into Equation 1:
(B =) (C + 3) - C = -1
3 = -1
Once again, this equation is not possible. So, we need to make another assumption.
Let's assume B = C + 4 and substitute it into Equation 1:
(B =) (C + 4) - C = -1
4 = -1
This equation is not possible.
It seems we have exhausted the assumption values B = C + 1, B = C + 2, B = C + 3, and B = C + 4 without finding a solution.
Therefore, there might be an error in the given information regarding the weights of the pairs of boxes.