A class took two tests , one in Physics and other in Maths `((1)/(3))^(rd)` class failed in Maths and `(1)/(2)` class failed in Physics . 60% of thoese passed in Physics also passed in Maths . If 300 students took both the tests then how many students failed in both ?

Answer

To solve the problem step by step, we can use the information provided in the question.

### Step 1: Determine the number of students who failed in each subject.
- Total students = 300
- Students who failed in Maths = \( \frac{1}{3} \) of 300 = \( 100 \)
- Students who failed in Physics = \( \frac{1}{2} \) of 300 = \( 150 \)

### Step 2: Calculate the number of students who passed in each subject.
- Students who passed in Maths = Total students - Students who failed in Maths
\[ = 300 - 100 = 200 \]
- Students who passed in Physics = Total students - Students who failed in Physics
\[ = 300 - 150 = 150 \]

### Step 3: Determine the number of students who passed in both subjects.
- According to the problem, 60% of those who passed in Physics also passed in Maths.
- Students who passed in Physics = 150
- Students who passed in both subjects = \( 60\% \) of 150 = \( 0.6 \times 150 = 90 \)

### Step 4: Use the principle of inclusion-exclusion to find the number of students who failed in both subjects.
- Let \( A \) be the set of students who passed in Maths, and \( B \) be the set of students who passed in Physics.
- We know:
- \( |A| = 200 \) (students who passed in Maths)
- \( |B| = 150 \) (students who passed in Physics)
- \( |A \cap B| = 90 \) (students who passed in both subjects)

Using the formula for the union of two sets:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Substituting the values:
\[ |A \cup B| = 200 + 150 - 90 = 260 \]

### Step 5: Calculate the number of students who failed in both subjects.
- Students who failed in both subjects = Total students - Students who passed in at least one subject
\[ = 300 - |A \cup B| = 300 - 260 = 40 \]

### Final Answer:
The number of students who failed in both subjects is **40**.

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