24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Rs. 200
Understanding the Average Contribution Problem
This problem involves calculating the contribution made by an individual (the teacher) based on changes in the average contribution of a group. The key concept here is the relationship between the average, the total sum, and the number of items or contributors.
The formula for the average is:
\[ \text{Average} = \frac{\text{Total Contribution}}{\text{Number of Contributors}} \]
From this, we can derive the formula for the total contribution:
\[ \text{Total Contribution} = \text{Average} \times \text{Number of Contributors} \]
Step-by-Step Calculation of Teacher's Contribution
Let's break down the problem into two stages: before the teacher contributes and after the teacher contributes.
Stage 1: Contribution by Students Only
- Number of students = 24
- Average contribution by students = Rs. 50
Using the formula for total contribution, we can find the total money collected by the 24 students:
\[ \text{Total contribution by students} = \text{Average contribution by students} \times \text{Number of students} \]
\[ \text{Total contribution by students} = 50 \times 24 \]
\[ \text{Total contribution by students} = 1200 \text{ Rs.} \]
So, the initial total contribution from the 24 students was Rs. 1200.
Stage 2: Contribution by Students and Teacher
- Number of students = 24
- Number of teacher = 1
- Total number of contributors = Number of students + Number of teacher = 24 + 1 = 25
- New average contribution (students + teacher) = Rs. 56
Now, let's find the new total contribution from all 25 contributors (students and teacher):
\[ \text{Total contribution (students + teacher)} = \text{New average contribution} \times \text{Total number of contributors} \]
\[ \text{Total contribution (students + teacher)} = 56 \times 25 \]
To calculate \(56 \times 25\):
\[ 56 \times 25 = 56 \times \frac{100}{4} = \frac{5600}{4} = 1400 \]
So, the new total contribution after the teacher contributed is Rs. 1400.
Calculating the Teacher's Contribution
The difference between the new total contribution and the initial total contribution by students is the teacher's contribution.
\[ \text{Teacher's contribution} = \text{Total contribution (students + teacher)} - \text{Total contribution by students} \]
\[ \text{Teacher's contribution} = 1400 - 1200 \]
\[ \text{Teacher's contribution} = 200 \text{ Rs.} \]
Therefore, the teacher's contribution is Rs. 200.
Summary of Teacher's Contribution Calculation
Here is a quick summary of the steps:
- Calculate the total contribution by the initial group of students: \(24 \text{ students} \times 50 \text{ Rs/student} = 1200 \text{ Rs}\).
- Determine the new number of contributors after the teacher joins: \(24 \text{ students} + 1 \text{ teacher} = 25 \text{ contributors}\).
- Calculate the new total contribution for the entire group: \(25 \text{ contributors} \times 56 \text{ Rs/contributor} = 1400 \text{ Rs}\).
- Subtract the initial total contribution from the new total contribution to find the teacher's contribution: \(1400 \text{ Rs} - 1200 \text{ Rs} = 200 \text{ Rs}\).
The teacher contributed Rs. 200.
Revision Table: Average Contribution Concepts
| Concept | Formula | Explanation |
|---|---|---|
| Average | \(\frac{\text{Sum of Quantities}}{\text{Number of Quantities}}\) | The sum of all values divided by the count of values. |
| Total Sum | \(\text{Average} \times \text{Number of Quantities}\) | The total value obtained by multiplying the average by the count. |
| Change in Total Sum | New Total - Old Total | The difference in the total value after a change occurs. |
Additional Information: Average Problems
Average problems often involve finding a missing value when the average of a group changes due to the addition or removal of an item or person. The key is always to work with the total sum. By finding the total sum before and after the change, you can easily determine the value of the item or person that caused the change. This approach is versatile and can be applied to various scenarios, such as finding a new score needed to achieve a certain average, or finding the value of a removed item.
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