Understanding Average Weight Problems

This problem involves calculating the weight of a new person joining a group when one person leaves, causing a change in the group's average weight. The key concept is that the total change in weight of the group is directly related to the change in the average weight.

Analyzing the Given Information

• Initial number of persons = 12
• Weight of the person who left = 58 kg
• Increase in the average weight of the remaining (new) group = 3.5 kg
• The number of persons in the group remains 12 after the change (one leaves, one joins).

Calculating the Total Increase in Weight

When the average weight of a group increases by a certain amount, the total weight of the group increases by that amount multiplied by the number of persons in the group.

In this case, the average weight of the 12 persons increases by 3.5 kg. So, the total increase in the weight of the group is:

\(\text{Total increase in weight} = \text{Increase in average weight} \times \text{Number of persons}\)

\(\text{Total increase in weight} = 3.5 \text{ kg/person} \times 12 \text{ persons}\)

Let's calculate this value:

\(3.5 \times 12 = (3 + 0.5) \times 12 = 3 \times 12 + 0.5 \times 12 = 36 + 6 = 42\)

So, the total increase in weight of the group is 42 kg.

Relating Total Increase to the New Person's Weight

The total weight of the group increased by 42 kg. This increase happened because a person weighing 58 kg was replaced by a new person. The difference between the new person's weight and the old person's weight accounts for this total increase.

Let \(W_{\text{new}}\) be the weight of the new person and \(W_{\text{old}}\) be the weight of the person who left.

\(\text{Total increase in weight} = W_{\text{new}} - W_{\text{old}}\)

We know the total increase in weight is 42 kg and \(W_{\text{old}}\) is 58 kg.

\(42 \text{ kg} = W_{\text{new}} - 58 \text{ kg}\)

Calculating the Weight of the New Person

To find the weight of the new person, we rearrange the equation:

\(W_{\text{new}} = \text{Total increase in weight} + W_{\text{old}}\)

\(W_{\text{new}} = 42 \text{ kg} + 58 \text{ kg}\)

\(W_{\text{new}} = 100 \text{ kg}\)

Therefore, the weight of the new person is 100 kg.

Summary of Calculation

The weight of the new person is 100 kg.

Revision Table: Average Weight Concepts

Additional Information: Handling Average Problems

Problems involving averages often require understanding the relationship between the average, the sum of values, and the number of items. When an item is added or removed, or when one item replaces another, the total sum of values changes, which in turn affects the average.

• Adding a person: If a new person joins a group, the total weight increases by the new person's weight, and the number of persons increases by one.
• Removing a person: If a person leaves a group, the total weight decreases by the person's weight, and the number of persons decreases by one.
• Replacement: When one person is replaced by another, the number of persons remains the same. The total weight changes by the difference between the new person's weight and the replaced person's weight. If the average increases, the new person is heavier than the old one. If the average decreases, the new person is lighter.

This particular problem is a case of replacement where the average increased, indicating the new person was heavier than the one replaced.