In an examination a candidate had to sit for three papers A, B, and C. The candidate secured 75% marks in Paper A, 80% marks in Paper B, and 60% marks in Paper C. If the weightage assigned to Papers A, B, and C were 40%, 50% and 10%, respectively, then find the weighted percentage of marks obtained by the candidate, when all the three papers were taken together.
76%
Understanding Weighted Percentage Marks
This problem asks us to calculate the overall percentage of marks obtained by a candidate in an examination, considering the different weightages assigned to each paper. This is a classic example of calculating a weighted average.
Calculating Weighted Contribution
To find the weighted percentage, we need to calculate the contribution of marks from each paper based on its assigned weightage. The formula for weighted average is:
$\text{Weighted Average} = \frac{\sum (\text{Value}_i \times \text{Weight}_i)}{\sum \text{Weight}_i}$
In this case, the values are the percentage marks obtained in each paper, and the weights are the percentage weightage assigned to each paper. The total weight is $40\% + 50\% + 10\% = 100\%$.
Let's calculate the weighted contribution for each paper:
- Paper A: The candidate secured 75% marks, and Paper A has a weightage of 40%.
Weighted contribution from Paper A = $75\% \times 40\%$ - Paper B: The candidate secured 80% marks, and Paper B has a weightage of 50%.
Weighted contribution from Paper B = $80\% \times 50\%$ - Paper C: The candidate secured 60% marks, and Paper C has a weightage of 10%.
Weighted contribution from Paper C = $60\% \times 10\%$
Step-by-Step Calculation
Let's convert the percentages to decimals for calculation or perform the calculation directly:
- Weighted contribution from Paper A = $75 \times \frac{40}{100} = 75 \times 0.40 = 30$
- Weighted contribution from Paper B = $80 \times \frac{50}{100} = 80 \times 0.50 = 40$
- Weighted contribution from Paper C = $60 \times \frac{10}{100} = 60 \times 0.10 = 6$
Alternatively, using percentages:
- Weighted contribution from Paper A = $\frac{75}{100} \times \frac{40}{100} \times 100\% = \frac{3000}{100} \% = 30\%$
- Weighted contribution from Paper B = $\frac{80}{100} \times \frac{50}{100} \times 100\% = \frac{4000}{100} \% = 40\%$
- Weighted contribution from Paper C = $\frac{60}{100} \times \frac{10}{100} \times 100\% = \frac{600}{100} \% = 6\%$
Now, we sum the weighted contributions:
Total weighted sum = $30 + 40 + 6 = 76$ (or $30\% + 40\% + 6\% = 76\%$)
Since the total weight is 100%, the weighted percentage of marks is simply the total weighted sum divided by the total weight (which is 1 in decimal form or 100%):
Weighted percentage of marks = $\frac{\text{Total weighted sum}}{\text{Total weight}} = \frac{76}{100} \times 100\% = 76\%$
Let's summarize the data and calculation in a table:
| Paper | Marks Secured (%) | Weightage (%) | Weighted Contribution |
|---|---|---|---|
| A | 75 | 40 | $75 \times \frac{40}{100} = 30$ |
| B | 80 | 50 | $80 \times \frac{50}{100} = 40$ |
| C | 60 | 10 | $60 \times \frac{10}{100} = 6$ |
| Total | 100 | $30 + 40 + 6 = 76$ |
The weighted percentage of marks obtained by the candidate when all three papers were taken together is 76%.
Revision Table: Key Concepts in Weighted Average
| Concept | Description | Formula |
|---|---|---|
| Weighted Average | An average where some values contribute more than others, determined by their weightages. | $\frac{\sum (x_i w_i)}{\sum w_i}$ (where $x_i$ are values, $w_i$ are weights) |
| Weightage | A measure of the importance or contribution of each data point to the total average. | Expressed as percentage, fraction, or ratio. Sum of weights often equals 1 or 100%. |
| Percentage Marks | Marks obtained out of a total, expressed as a percentage. | $\frac{\text{Marks Obtained}}{\text{Total Marks}} \times 100\%$ |
Additional Information: Applications of Weighted Averages
Weighted averages are used in many real-world situations, such as:
- Calculating GPA: Different courses have different credit hours (weights).
- Financial Indexes: Stock prices are weighted by market capitalization.
- Manufacturing: Averaging quality control data from different production lines with varying volumes.
- Surveys and Statistics: Adjusting data to reflect population demographics.
Understanding weighted averages is important for correctly interpreting data where different components have varying levels of significance.
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