In a class of 180 students 95 passed in History, 38 failed in Geography.  73 passed in both.

To analyze the situation involving the students and their performance in History and Geography, we can use set theory and Venn diagrams. We'll denote the following:

• Let $H$ be the set of students who passed in History.
• Let $G$ be the set of students who passed in Geography.

Given:

• Total students, $N = 180$
• Students who passed in History, $|H| = 95$
• Students who failed in Geography, $|G'| = 38$
• Students who passed in both subjects, $|H \cap G| = 73$

1. ### Break Down the Problem

We need to determine:

1. The number of students who passed in Geography.
2. The number of students who failed in History.
3. The number of students who passed in either subject.

2. ### Relevant Concepts

We use the following formulas from set theory:

• The total number of students who passed in either subject: $|H \cup G| = |H| + |G| - |H \cap G|$

• The number of students who failed in History can be calculated as: $\text{Failing in History} = N - |H|$

3. ### Analysis and Detail

Step 1: Find $|G|$ (Students who passed in Geography)

Since $|G'| = 38$, the number of students who passed in Geography is: $|G| = N - |G'| = 180 - 38 = 142$

Step 2: Calculate $|H \cup G|$

Using the formula for the union of two sets: $|H \cup G| = |H| + |G| - |H \cap G| = 95 + 142 - 73 = 164$

Step 3: Find the number of students who failed in History

$\text{Failing in History} = N - |H| = 180 - 95 = 85$

4. ### Verify and Summarize

• Students passing Geography: $|G| = 142$
• Students passing either subject: $|H \cup G| = 164$
• Students failing in History: $85$

Final Answer

• Total students who passed in Geography: $142$
• Total students who passed in either History or Geography: $164$
• Total students who failed in History: $85$