< b="" />Understanding the Problem< />

We are given that 80% of students passed in English, 85% of students passed in Math, and 75% of students passed in both subjects. We are also told that 40 students failed in both subjects. We need to determine the total number of students.

< b="" />Key Points< />

Let's identify the key points given in the problem:

- 80% of students passed in English
- 85% of students passed in Math
- 75% of students passed in both subjects
- 40 students failed in both subjects

< b="" />Solution Approach< />

To find the total number of students, we can use the concept of set theory and Venn diagrams. Let's represent the three sets of students as follows:

- Set A represents students who passed in English.
- Set B represents students who passed in Math.
- Set C represents students who passed in both subjects.

< b="" />Using Venn Diagrams< />

Now, let's represent the given information in a Venn diagram:

- 80% of students passed in English, which means 20% of students failed in English. This can be represented as (100% - 80%) = 20%.
- 85% of students passed in Math, which means 15% of students failed in Math. This can be represented as (100% - 85%) = 15%.
- 75% of students passed in both subjects. This can be represented as the overlapping region of sets A and B.

< b="" />Calculating the Number of Students< />

From the Venn diagram, we can see that the number of students who failed in English (20%) is the sum of students who failed in both subjects and students who only failed in English. Similarly, the number of students who failed in Math (15%) is the sum of students who failed in both subjects and students who only failed in Math.

Let's assume the total number of students is 'x'. We can calculate the number of students who failed in both subjects using the formula:

(Number of students who failed in both subjects) = (Percentage of students who failed in English) + (Percentage of students who failed in Math) - (Percentage of students who failed in both subjects)

Substituting the given values:

40 = 20% + 15% - 75%

Simplifying the equation:

40 = 35% - 75%

Adding 75% to both sides:

40 + 75% = 35%

Dividing both sides by 35:

115% = 40

Dividing both sides by 115:

1% = 40/115

Multiplying both sides by 100 to find the total number of students:

100% = (40/115) x 100

≈ 34.78

Therefore, the total number of students is approximately 34.78. However, since we cannot have fractional students, we need to round up to the nearest whole number.

Hence, the total number of students is 35.

< b="" />Conclusion< />

The total number of students is 35, not 400 as mentioned in option B. It seems there may be an error in the options provided.