In a group of 600 students who take tests in both physics and chemistry, 35 percent fail in physics and 45 percent fail in chemistry. If 40 percent of those who pass in chemistry also pass in physics, how many students fail in both subjects?

Introduction / Context:
This is a Venn diagram or set theory style question involving two subjects, physics and chemistry. We are given percentages of failures in each subject and additional information about those who pass in one subject and also pass in the other. The goal is to find how many students fail in both subjects simultaneously. This type of question regularly appears in aptitude and entrance exams.

Given Data / Assumptions:

• Total number of students = 600.
• 35 percent fail in physics.
• 45 percent fail in chemistry.
• Therefore 65 percent pass in physics and 55 percent pass in chemistry.
• Forty percent of those who pass in chemistry also pass in physics.
• We must find the number of students who fail in both physics and chemistry.

Concept / Approach:
We can treat the sets as follows: P is the set of students who fail physics, C is the set of students who fail chemistry. Their complements are students who pass. We know the sizes of P and C from the given percentages. The key extra information relates to students who pass both subjects, because it tells us that 40 percent of those who pass chemistry also pass physics. Once we find the number who pass both, we can work backwards to figure out how many are left to fail both subjects.

Step-by-Step Solution:
Step 1: Number failing physics = 35 percent of 600 = 0.35 * 600 = 210.Step 2: Number failing chemistry = 45 percent of 600 = 0.45 * 600 = 270.Step 3: Number passing chemistry = 600 - 270 = 330.Step 4: Forty percent of those who pass chemistry also pass physics, so students passing both subjects = 0.40 * 330 = 132.Step 5: Let x be the number who fail both subjects. Then failing only physics is 210 - x and failing only chemistry is 270 - x.Step 6: Total students can be expressed as 600 = (pass both) + (fail only physics) + (fail only chemistry) + (fail both) = 132 + (210 - x) + (270 - x) + x.Step 7: Simplify the expression: 132 + 210 + 270 - x = 612 - x. Set 612 - x = 600, giving x = 12.

Verification / Alternative check:
We can verify by explicitly listing category sizes. Failing both subjects is 12. Failing only physics is 210 - 12 = 198. Failing only chemistry is 270 - 12 = 258. Passing both is 132. Summing these, 12 + 198 + 258 + 132 = 600, matching the total. Also, passing chemistry totals 132 + 198 = 330, and 40 percent of 330 is 132, which confirms the condition about passing both subjects. Everything is consistent.

Why Other Options Are Wrong:
A value of 60 or 138 or 162 or 180 for those who fail both would break at least one condition. For example, if 60 failed both, then passing chemistry and physics numbers would not match the 40 percent condition. Careful algebra shows that only x = 12 satisfies the equations that come from the given percentages and the relation between passes in the two subjects. Therefore, the other numerical options do not fit the data.

Common Pitfalls:
Many students mix up failing and passing sets or misinterpret the phrase 40 percent of those who pass in chemistry also pass in physics. Another common mistake is to try to use inclusion exclusion directly on the failing sets without incorporating the extra condition about passes. Drawing a Venn diagram or using variables for each region and writing equations is a reliable way to avoid confusion.

Final Answer:
The number of students who fail in both physics and chemistry is 12, so the correct option is 12.