In a class of 180 students, each student took three tests: Science, History, and Geography. The number of students who passed in the three tests were 90, 120, and 100 respectively. If none of the students failed in all three subjects and the number of students who passed in at least one subject was atmost 50% more than those who passed in at least two subjects, then maximum how many students passed in all three subjects?

Let's break down the problem step by step:

Step 1: Understand the problem The problem is asking for the maximum number of students who passed in all three subjects given certain conditions.

Step 2: Define the variables Let's denote:

• A as the number of students who passed in Science
• B as the number of students who passed in History
• C as the number of students who passed in Geography
• X as the number of students who passed in all three subjects
• Y as the number of students who passed in at least two subjects
• Z as the number of students who passed in at least one subject

From the problem, we know that A = 90, B = 120, C = 100, and Z ≤ 1.5Y.

Step 3: Formulate the equations We can formulate the following equations from the problem:

• A + B + C - Y = Z (since the students who passed in at least one subject are those who passed in Science, History, or Geography, minus those who passed in at least two subjects)
• Y ≥ X (since the students who passed in at least two subjects include those who passed in all three subjects)

Step 4: Substitute the known values Substituting the known values into the equations, we get:

• 90 + 120 + 100 - Y = Z
• Z ≤ 1.5Y

Step 5: Solve the equations To maximize X, we need to maximize Y. From the second equation, we can see that the maximum value of Y is when Z = 1.5Y. Substituting this into the first equation, we get:

• 90 + 120 + 100 - Y = 1.5Y
• 310 = 2.5Y
• Y = 124

Since Y ≥ X, the maximum number of students who passed in all three subjects is 124.