In an examination of a certain class, at least `70%` of the students failed in Physics, at least `72%` failed in Chemistry, at least `80%` failed in Mathematics and at least `85%` failed in English. How many at least must have failed in all the four subjects ?

Options

A. `5%`
B. `7%`
C. `15%`
D. Cannot be determined due to insufficient data

Answer

To solve the problem, we will use the principle of inclusion-exclusion and the information given about the percentages of students who failed in each subject.

### Step-by-Step Solution:

1. **Define Total Students**:
Let's assume the total number of students in the class is 100. This simplifies our calculations since we can directly use percentages as numbers.

2. **Calculate Students Passing Each Subject**:
- **Physics**: At least 70% failed, so at most 30% passed.
- **Chemistry**: At least 72% failed, so at most 28% passed.
- **Mathematics**: At least 80% failed, so at most 20% passed.
- **English**: At least 85% failed, so at most 15% passed.

3. **List the Maximum Passing Students**:
- Students passing Physics: 30
- Students passing Chemistry: 28
- Students passing Mathematics: 20
- Students passing English: 15

4. **Calculate Total Students Passing**:
Now we add the maximum number of students passing each subject:
\[ \text{Total Passing} = 30 + 28 + 20 + 15 = 93 \]

5. **Find Students Failing All Subjects**:
Since there are 100 students in total, the number of students who must have failed all four subjects can be calculated as:
\[ \text{Students Failing All} = \text{Total Students} - \text{Total Passing} = 100 - 93 = 7 \]

6. **Conclusion**:
Therefore, at least 7% of the students must have failed in all four subjects.

### Final Answer:
At least 7% of the students failed in all four subjects.

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