In a class of 180 students, each student took three tests: Science, History, and Geography. The number of students who passed in the three tests were 90, 120, and 100 respectively. If
none of the students failed in all three subjects and the number of students who passed in at least one subject was atmost 50% more than those who passed in at least two subjects, then
maximum how man students assed in all three subjects?

To solve this, let's use the principle of inclusion-exclusion and denote the following variables:

- Let:
  - \( S \): Students who passed Science.
  - \( H \): Students who passed History.
  - \( G \): Students who passed Geography.
- We know:
  - \( |S| = 90 \)
  - \( |H| = 120 \)
  - \( |G| = 100 \)

Given that none of the students failed in all subjects, we have:

1. **The total number of students, \( |S \cup H \cup G| \), is 180.**
2. **The number of students who passed at least one subject is at most 50% more than those who passed in at least two subjects.**

Let’s set up our problem with variables and calculate:

- Let \( x \) be the number of students who passed all three subjects, \( |S \cap H \cap G| \).
- Let \( |S \cap H| = a \), \( |S \cap G| = b \), and \( |H \cap G| = c \).

### Step 1: Inclusion-Exclusion Principle
Using the inclusion-exclusion principle:

\[
|S \cup H \cup G| = |S| + |H| + |G| - (|S \cap H| + |S \cap G| + |H \cap G|) + |S \cap H \cap G|
\]

Substitute known values:

\[
180 = 90 + 120 + 100 - (a + b + c) + x
\]

Simplifying:

\[
180 = 310 - (a + b + c) + x
\]

\[
a + b + c - x = 130
\]

### Step 2: Condition on Students Passing at Least Two Subjects
The number of students who passed **at least two subjects** is given by \( (a + b + c - 3x) \).

We are told that the number of students who passed at least one subject is **at most 50% more** than the number who passed at least two subjects. Therefore:

\[
|S \cup H \cup G| \leq 1.5 \times (a + b + c - 3x)
\]

Substituting \( |S \cup H \cup G| = 180 \):

\[
180 \leq 1.5 \times (a + b + c - 3x)
\]

Now, we will solve this system of equations to determine the maximum value of \( x \). 

Continuing from above, let's refine our approach for maximizing \( x \), the number of students who passed all three subjects. 

We have:

1. **From the inclusion-exclusion principle**:
   \[
   a + b + c - x = 130
   \]

2. **Condition on students passing at least two subjects**:
   \[
   180 \leq 1.5 \times (a + b + c - 3x)
   \]

### Step 3: Substitute \( a + b + c \)
From the first equation, we can express \( a + b + c \) in terms of \( x \):
\[
a + b + c = 130 + x
\]

Substitute this into the inequality from the condition on students passing at least two subjects:
\[
180 \leq 1.5 \times (130 + x - 3x)
\]

Simplify the inequality:
\[
180 \leq 1.5 \times (130 - 2x)
\]

\[
180 \leq 195 - 3x
\]

Rearrange to isolate \( x \):
\[
3x \leq 15
\]

\[
x \leq 5
\]

### Conclusion
The maximum number of students who passed all three subjects is therefore \( \boxed{5} \). 

---

Would you like more details or have any questions? Here are some related questions to consider:

1. How does the inclusion-exclusion principle work in set theory?
2. How can we apply this method to similar problems with different numbers of sets?
3. What if some students could have failed all three subjects? How would that change the approach?
4. How would changing the "50% more" condition impact the answer?
5. Can we derive a formula for problems involving more than three subjects?

**Tip:** For problems with conditions involving multiple overlapping groups, the inclusion-exclusion principle is a powerful tool for managing over-counting.

In a class of 180 students, each student took three tests: Science, History, and Geography. The number of students who passed in the three tests were 90, 120, and 100 respectively. If none of the students failed in all three subjects and the number of students who passed in at least one subject was at most 50% more than those who passed in at least two subjects, then maximum how many students passed in all three subjects?

This solution guides you through finding the maximum number of students who passed all three subjects using the inclusion-exclusion principle. Given 180 students, with 90 passing Science, 120 History, and 100 Geography, the solution determines the count of students passing all subjects while applying algebraic constraints.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation