In an examination 70% candidates passed in history and 50% in geography and 20% students failed in both the subjects . If 500 students passed in both subjects , then how many students appeared in the exAMINATION ?

Let's use set theory to solve this problem, considering the students who passed History and Geography as two sets.

- Let the total number of students be \( N \).
- 70% passed in History, so the number of students who passed in History is \( 0.7N \).
- 50% passed in Geography, so the number of students who passed in Geography is \( 0.5N \).
- 20% failed in both subjects, so 80% of the students passed at least one subject, which means \( 0.8N \) passed in at least one subject.
- The number of students who passed both subjects is given as 500.

We know from set theory that:

\[
\text{Number of students who passed in at least one subject} = (\text{Number of students who passed History}) + (\text{Number of students who passed Geography}) - (\text{Number of students who passed both History and Geography})
\]

Substituting the known values:

\[
0.8N = 0.7N + 0.5N - 500
\]

Simplifying:

\[
0.8N = 1.2N - 500
\]

Solving for \( N \):

\[
0.8N - 1.2N = -500
\]

\[
-0.4N = -500
\]

\[
N = \frac{500}{0.4} = 1250
\]

Thus, the total number of students who appeared in the examination is **1250**.

Would you like further clarification or additional details?

Here are 5 related questions:
1. What is the minimum number of students who passed exactly one subject?
2. How many students passed only History?
3. How many students passed only Geography?
4. What percentage of students passed both subjects?
5. What percentage of students passed at least one subject?

**Tip:** Always keep an eye on double-counting when dealing with union and intersection of sets!

In an examination 70% candidates passed in history and 50% in geography and 20% students failed in both the subjects. If 500 students passed in both subjects, then how many students appeared in the examination?

This problem involves calculating the total number of students who appeared in an examination using set theory. With 70% passing in History, 50% passing in Geography, and 20% failing both subjects, learn how to find the total number of students using the union and intersection of sets. Suitable for students in Grades 9-10.

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