The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
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Calculating the New Average Weight of the Class
This problem asks us to find the new average weight of a class after some students leave and a different group of students join. To solve this, we first need to find the initial total weight of the class, then adjust it based on the students leaving and joining, and finally calculate the new average using the new total weight and the new number of students.
Step 1: Calculate the Initial Total Weight
We are given the initial number of students and their average weight. The total weight is the product of the number of students and their average weight.
- Initial number of students = 49
- Initial average weight = 39 kg
- Initial total weight = Number of students $\times$ Average weight
- Initial total weight = $49 \times 39$ kg
Let's calculate the initial total weight:
$\qquad 49 \times 39 = 1911$ kg
So, the initial total weight of the 49 students is 1911 kg.
Step 2: Calculate the Total Weight of Students Leaving
Seven students leave the class, and their average weight is 40 kg. We calculate their total weight.
- Number of students leaving = 7
- Average weight of students leaving = 40 kg
- Total weight of students leaving = Number of students leaving $\times$ Average weight of students leaving
- Total weight of students leaving = $7 \times 40$ kg
Let's calculate the total weight of students leaving:
$\qquad 7 \times 40 = 280$ kg
The total weight removed from the class is 280 kg.
Step 3: Calculate the Total Weight of Students Joining
Another seven students join the class, and their average weight is 54 kg. We calculate their total weight.
- Number of students joining = 7
- Average weight of students joining = 54 kg
- Total weight of students joining = Number of students joining $\times$ Average weight of students joining
- Total weight of students joining = $7 \times 54$ kg
Let's calculate the total weight of students joining:
$\qquad 7 \times 54 = 378$ kg
The total weight added to the class is 378 kg.
Step 4: Calculate the New Total Weight of the Class
The new total weight is the initial total weight minus the weight of students who left plus the weight of students who joined.
- New total weight = Initial total weight - Total weight of students leaving + Total weight of students joining
- New total weight = $1911$ kg - $280$ kg + $378$ kg
Let's perform the calculation:
$\qquad 1911 - 280 = 1631$ kg
$\qquad 1631 + 378 = 2009$ kg
The new total weight of the class is 2009 kg.
Step 5: Determine the New Number of Students
The initial number of students was 49. Seven students left, and seven new students joined. The change in the number of students is $7 - 7 = 0$.
- New number of students = Initial number of students - Students leaving + Students joining
- New number of students = $49 - 7 + 7 = 49$
The number of students in the class remains 49.
Step 6: Calculate the New Average Weight
The new average weight is the new total weight divided by the new number of students.
- New average weight = New total weight / New number of students
- New average weight = $2009$ kg / $49$
Let's perform the division:
$\qquad \frac{2009}{49}$
We can estimate or perform long division:
$2009 \div 49 = 41$
The new average weight of the class is 41 kg.
Let's summarize the steps and results in a table:
| Description | Number of Students | Average Weight (kg) | Total Weight (kg) |
|---|---|---|---|
| Initial Class | 49 | 39 | $49 \times 39 = 1911$ |
| Students Leaving | 7 | 40 | $7 \times 40 = 280$ |
| Students Joining | 7 | 54 | $7 \times 54 = 378$ |
| New Class | $49 - 7 + 7 = 49$ | ? | $1911 - 280 + 378 = 2009$ |
New Average Weight = $\frac{\text{New Total Weight}}{\text{New Number of Students}} = \frac{2009}{49} = 41$ kg.
Conclusion
The new average weight of the class is 41 kg.
Revision Table: Average Weight Calculation
Understanding how average weight changes when students leave and join is a common problem in calculating averages. Here's a quick look at the key concepts:
| Concept | Formula | Application |
|---|---|---|
| Average | $\text{Average} = \frac{\text{Sum of values}}{\text{Number of values}}$ | Used to find average weight or number of items. |
| Total Sum | $\text{Total Sum} = \text{Average} \times \text{Number of values}$ | Used to find total weight given average and number of students. |
| Change in Total Sum | $\text{Change} = \text{Sum Added} - \text{Sum Removed}$ | Used to find the net change in total weight. |
| New Average | $\text{New Average} = \frac{\text{Initial Total Sum} + \text{Change in Total Sum}}{\text{New Number of values}}$ | Used to find the final average after changes. |
Additional Information: Understanding Averages
The average, also known as the mean, is a fundamental concept in statistics. It represents a typical value in a set of data. When dealing with groups leaving or joining, the key is always to work with the total sum (or total weight in this case) because averages cannot be simply added or subtracted directly unless the number of items is the same.
- Why not average the averages? You cannot simply average the initial average (39 kg), the leaving average (40 kg), and the joining average (54 kg). Averages are weighted by the number of items they represent. A group of 7 students with an average weight of 54 kg contributes more to the total weight than a group of 7 students with an average weight of 40 kg, even though the group sizes are the same. If the group sizes were different (e.g., 10 left and 5 joined), it would be even more critical to use the total weight method.
- Impact of changes: When a group with a higher average leaves than the group joining, the overall class average tends to decrease. Conversely, when a group with a higher average joins than the group leaving (as in this problem, 54 kg > 40 kg), the overall class average tends to increase.
- Applications: Calculating averages and tracking changes is important in many areas, such as finance (average stock price), demographics (average age), and science (average temperature).
Mastering the calculation of total sum from the average is crucial for solving problems involving changes in groups.
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