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 find the minimum percentage of students who failed in all four subjects, we can use the principle of inclusion-exclusion. Let's denote:

- \( P \): Percentage of students who failed in Physics = 70%
- \( C \): Percentage of students who failed in Chemistry = 72%
- \( M \): Percentage of students who failed in Mathematics = 80%
- \( E \): Percentage of students who failed in English = 85%

We want to find the minimum percentage of students who failed in all four subjects, denoted as \( x \).

### Step 1: Calculate the total percentage of students who failed in at least one subject.

Using the principle of inclusion-exclusion, we can express the percentage of students who failed in at least one subject as follows:

\[ \text{Percentage who failed in at least one subject} = P + C + M + E - (A + B + C + D) \]

Where \( A, B, C, D \) are the overlaps of students failing in pairs of subjects. Since we do not have specific data on these overlaps, we can assume the worst-case scenario where the overlaps are minimized.

### Step 2: Calculate the total percentage of students who passed in at least one subject.

The percentage of students who passed in at least one subject is given by:

\[ \text{Percentage who passed in at least one subject} = 100\% - \text{Percentage who failed in at least one subject} \]

### Step 3: Calculate the minimum percentage of students who failed in all subjects.

To find the minimum percentage of students who failed in all subjects, we can use the following formula:

\[ \text{Percentage who failed in all subjects} = P + C + M + E - 300\% \]

Substituting the values:

\[ \text{Percentage who failed in all subjects} = 70\% + 72\% + 80\% + 85\% - 300\% \]

Calculating this gives:

\[ \text{Percentage who failed in all subjects} = 307\% - 300\% = 7\% \]

### Conclusion:

Thus, at least **7%** of the students must have failed in all four subjects.

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