Question

The average weight of 55 students is \(55 \mathrm{~kg}\) and the average weight of another 45 students is \(45 \mathrm{~kg}\). Find the average weight of all the students. (1) \(48 \mathrm{~kg}\) (2) \(50 \mathrm{~kg}\) (3) \(50.5 \mathrm{~kg}\) (4) \(52.25 \mathrm{~kg}\)

Short answer

Answer: (3) 50.5 kg

Step-by-step solution

Calculate the combined weight of each group

First, we need to find the combined weight of the students in each group. We can do this by multiplying the average weight with the number of students in each group. For the first group of 55 students with an average weight of 55 kg, the combined weight will be: \(55 \text{ students} \times 55 \mathrm{~kg} = 3025 \mathrm{~kg}\) For the second group of 45 students with an average weight of 45 kg, the combined weight will be: \(45 \text{ students} \times 45 \mathrm{~kg} = 2025 \mathrm{~kg}\)

Find the total combined weight and the number of students

Now, we will find the total combined weight of all students and the total number of students. Total combined weight = Combined weight of the first group + Combined weight of the second group \(= 3025 \mathrm{~kg} + 2025 \mathrm{~kg} = 5050 \mathrm{~kg}\) Total number of students = Number of students in the first group + Number of students in the second group \(= 55 \text{ students} + 45 \text{ students} = 100 \text{ students}\)

Calculate the overall average weight

Finally, let's calculate the overall average weight by dividing the total combined weight by the total number of students. Overall average weight = \(\frac{\text{Total combined weight}}{\text{Total number of students}}\) \(= \frac{5050 \mathrm{~kg}}{100 \text{ students}} = 50.50 \mathrm{~kg}\) So the correct answer is: (3) \(50.5 \mathrm{~kg}\)

Key concepts

Weighted Average

The concept of a weighted average is a key principle when evaluating the average of different groups with varying sizes. When calculating the weighted average, each group's average is given a "weight," which depends on the size of the group. In this exercise, we have two groups of students with different average weights and sizes.

• The first group has 55 students with an average weight of 55 kg.
• The second group has 45 students with an average weight of 45 kg.

To determine the overall average, we consider the "weight" of each group's average by multiplying the average by the number of students in that group. By doing this, we are effectively scaling the average to reflect its impact based on group size, thereby providing a more accurate overall average. Thus, the weighted average is specifically helpful in cases where the groups being considered aren't equal in size, as it balances each group's contribution to reflect their importance.

Arithmetic Mean

The arithmetic mean, often simply referred to as the average, is the sum of a set of values divided by the number of values. This concept is used frequently in everyday math to find a single value representing a set of numbers. In this problem, we apply the concept of the arithmetic mean to find the average weight of all students combined.
To compute the arithmetic mean:

• Add up the total weight of all students (both groups).
• Divide this total by the total number of students.

The computation for this exercise would be as follows:
- Total combined weight = 5050 kg,- Total number of students = 100.
Therefore, the arithmetic mean (average weight) is calculated as:
\[ \text{Overall average weight} = \frac{5050 \text{ kg}}{100 \text{ students}} = 50.5 \text{ kg} \]This output gives us the single value that represents the central tendency of the data of all 100 students' weights.

Combined Weight Calculation

Understanding how to calculate combined weight is essential in solving problems involving groups with different characteristics. In this exercise, we're looking to consolidate weights from two different groups before calculating a final average.
First, let's estimate the weight of each group by multiplying the group's average weight by the number of students in that group. Here are the maths for this exercise:

• For the first group: 55 students \( \times \) 55 kg = 3025 kg.
• For the second group: 45 students \( \times \) 45 kg = 2025 kg.

By adding these combined weights, we determine the total weight for all students:- Total combined weight = 3025 kg + 2025 kg = 5050 kg.
In accomplishing these steps, combining weights helps in adjusting the scale to suit the entire population rather than focusing on only one segment. Subsequent to the calculation of combined weight, it becomes straightforward to find out the overall average by dividing by the total number of students.