In an examination, 70% of the students passed in the Paper I, and 60% of the students passed in the Paper II: 15% of the students failed in both the papers while 270 students passed in both the papers. What is the total number of students?

To find the total number of students, we use the principle of set theory.

Step 1: Determine the percentage of students who passed at least one subject.
Since 15% of students failed in both papers, the percentage of students who passed at least one paper is:
100% - 15% = 85%

Step 2: Calculate the percentage of students who passed in both papers.
Let P(I) be those passing Paper I and P(II) be those passing Paper II.
Using the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
85% = 70% + 60% - n(Both)
85% = 130% - n(Both)
n(Both) = 130% - 85% = 45%

Step 3: Calculate the total number of students.
We are given that 270 students passed in both papers, which represents 45% of the total.
If T is the total number of students:
45% of T = 270
T = (270 × 100) / 45
T = 600