In an examination, 92% of the students passed and 480 students failed. If so, how many students appeared in the examination?
6000
Understanding the Examination Result Problem
This question asks us to find the total number of students who participated in an examination. We are given the percentage of students who passed and the specific number of students who failed. To solve this, we first need to figure out what percentage of students failed the examination.
Calculating the Percentage of Failed Students
In any examination, the total percentage of students appearing is 100%. Students either pass or fail. Therefore, the percentage of students who failed is the total percentage minus the percentage of students who passed.
- Total percentage of students = 100%
- Percentage of students who passed = 92%
- Percentage of students who failed = Total percentage - Percentage of students who passed
- Percentage of students who failed = 100% - 92% = 8%
So, 8% of the students who appeared for the examination failed.
Relating Percentage Failed to Number Failed
We are given that 480 students failed the examination. We now know that this number represents 8% of the total number of students who appeared.
Let 'T' be the total number of students who appeared in the examination.
According to the problem:
8% of T = 480
Solving for the Total Number of Students
To find the total number of students (T), we can write the percentage as a fraction:
$\frac{8}{100} \times T = 480$
Now, we need to isolate T. We can do this by multiplying both sides of the equation by $\frac{100}{8}$:
$T = 480 \times \frac{100}{8}$
We can simplify the calculation by dividing 480 by 8:
$480 \div 8 = 60$
So, the equation becomes:
$T = 60 \times 100$
$T = 6000$
Therefore, the total number of students who appeared in the examination was 6000.
Verification of the Solution
Let's check if our answer is correct. If there were 6000 students in total, and 8% failed, the number of failed students would be:
$8\% \text{ of } 6000 = \frac{8}{100} \times 6000 = 8 \times 60 = 480$
This matches the number of failed students given in the question (480). Also, 92% passed:
$92\% \text{ of } 6000 = \frac{92}{100} \times 6000 = 92 \times 60 = 5520$
Total students = Passed students + Failed students = $5520 + 480 = 6000$. This confirms our total is correct.
| Category | Percentage | Number of Students |
|---|---|---|
| Passed | 92% | $92\% \text{ of } 6000 = 5520$ |
| Failed | 8% (100% - 92%) | 480 (Given) |
| Total | 100% | $5520 + 480 = 6000$ |
The calculation confirms that if 6000 students appeared, 480 students failing corresponds to 8% failure rate, which is consistent with a 92% pass rate.
Revision Table: Examination Student Calculation
| Concept | Formula / Calculation | Value |
|---|---|---|
| Percentage Passed | Given | 92% |
| Number Failed | Given | 480 |
| Percentage Failed | 100% - Percentage Passed | $100\% - 92\% = 8\%$ |
| Total Students (T) | Percentage Failed of T = Number Failed | $\frac{8}{100} \times T = 480$ |
| Total Students (T) | $T = \frac{\text{Number Failed}}{\text{Percentage Failed}} \times 100$ | $T = \frac{480}{8} \times 100 = 60 \times 100 = 6000$ |
Additional Information: Percentage Calculations in Exams
Understanding percentages is crucial for solving many problems related to examinations and results. Here are some key points:
- Total is always 100%: When dealing with percentages of a whole (like students in an exam), the total represents 100%.
- Complementary Percentages: If students are divided into categories like 'passed' and 'failed', their percentages add up to 100%. For example, if 92% passed, $100\% - 92\% = 8\%$ must have failed.
- Finding the Whole from a Part: If you know the percentage that a certain number represents, you can find the total (the whole). The formula is:
$\text{Total} = \frac{\text{Part (Number)}}{\text{Percentage (in decimal or fraction)}} = \frac{\text{Part}}{\text{Percentage}} \times 100$ - Finding a Part from the Whole: If you know the total and a percentage, you can find the number that represents that percentage:
$\text{Part (Number)} = \text{Percentage (in decimal or fraction)} \times \text{Total} = \frac{\text{Percentage}}{100} \times \text{Total}$
These basic concepts help in tackling various percentage-based problems efficiently.
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