The sum of the weights of 8 children is 520 pounds. No two children have the same weight. What is the average(mean) weight of he fourth and fifth heaviest children if the mean and median weights of all children are equal?

mean is 520/8 = 65

so, the weights are

? ? ? (65 65) ? ? ?
the two middle weights add to 130. They are not both 65, but their average is.

To solve this problem, let's first understand what information we have:

- There are 8 children.
- The sum of their weights is 520 pounds.
- No two children have the same weight.
- The mean and median weights are equal.

Step 1: Find the median weight
Since there are 8 children, the median weight will be the weight of the 4th child, because the weights are ordered. The weights of the other children are ranked evenly on either side of the median.

Step 2: Find the mean weight
The mean weight is obtained by dividing the total weight by the number of children. In this case, the mean weight will also be the median weight because it is given that the mean and median weights are equal.

Step 3: Calculate the total weight
The total weight of the 8 children is given as 520 pounds.

Step 4: Find the sum of the weights of the four lightest children
To calculate the sum of the weights of the four lightest children, subtract the sum of the weights of the two heaviest children from the total weight. Since there are no two children with the same weight, the heaviest children are the 7th and 8th children.

Step 5: Calculate the sum of the weights of the fourth and fifth heaviest children
To calculate the sum of the weights of the fourth and fifth heaviest children, subtract the sum of the weights of the six lightest children from the total weight.

Step 6: Calculate the average weight
To calculate the average weight, divide the sum of the weights of the fourth and fifth heaviest children by 2 (since there are two children).

Step 7: Compute the answer:
By following the steps above, we can find the average (mean) weight of the fourth and fifth heaviest children.

To find the average weight of the fourth and fifth heaviest children, we first need to determine the median weight of all the children. Since the mean and median weights are equal, we can calculate the median weight by finding the weight that separates the lower half and upper half of the weights.

Given that there are 8 children, the median weight will be the weight of the 4th child (when arranged in ascending order). To find this weight, we need to arrange the weights in ascending order.

Let's assume the weights of the 8 children are:

w1, w2, w3, w4, w5, w6, w7, w8

Since no two children have the same weight, we can arrange the weights from least to greatest:

w1 < w2 < w3 < w4 < w5 < w6 < w7 < w8

The median weight will be w4, as it represents the middle value when the weights are arranged in ascending order.

Now, let's consider the sum of the weights of all the children. We know that the sum of the weights of 8 children is 520 pounds.

w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 = 520

Since the median weight is w4, we can remove it from the equation:

w1 + w2 + w3 + w5 + w6 + w7 + w8 = 520 - w4

Given that the mean weight is equal to the median weight, we can rewrite the equation as:

(w1 + w2 + w3 + w5 + w6 + w7 + w8) / 7 = w4

Simplifying the equation further, we get:

w1 + w2 + w3 + w5 + w6 + w7 + w8 = 7w4

Now, we have another equation with the sum of the weights. We can use these two equations to solve for both the median weight (w4) and the sum of the weights excluding the median weight.

However, without additional information or constraints, it is not possible to find unique values for the weights or the average weight of the fourth and fifth heaviest children. More information is needed to solve the problem definitively.