Question

In a small college, students are allowed to take only one specialization. Traditionally, only two specializations are offered: Science and Arts. Students enrolled to specialize in Science must take Physics and Mathematics subjects, while students enrolled to specialize in Arts must take Economics and Political Science subjects. Students enrolled in Science are not allowed to take either Economics or Political Science, while students enrolled in Arts are not allowed to take either Physics or Mathematics.
Recently, the college has started a third specialization called MatEco that requires students to take Economics and Mathematics. However, MatEco students would not be allowed to take either Physics or Political Science. When the college opens this new specialization for enrolment, it allows students, originally enrolled in Science or Arts, to switch to MatEco.
From among the students originally enrolled in Arts, 20 students switch to MatEco. This makes the number of Science students twice the number of Arts students. After this, from among the students who originally enrolled in Science, 45 students switch to MatEco. This makes the number of Arts students twice the number of Science students.
In total, how many students, from among those originally enrolled in Science or Arts, are now taking Economics?

• A. 45
• B. 65
• C. 80
• D. 95
• E. None of the remaining options is correct.

Answer 1

The Correct Option is D

To solve this problem, we need to assess the distribution of students across the specializations and the changes made due to the introduction of the MatEco specialization. Let's denote the initial number of students in each specialization as follows:

1. Let S be the number of students originally enrolled in Science.
2. Let A be the number of students originally enrolled in Arts.

After introducing MatEco:

1. 20 students switch from Arts to MatEco, leaving (A - 20) students in Arts.
2. This makes the number of Science students twice the number of Arts students: S = 2(A - 20).
3. 45 students switch from Science to MatEco, leaving (S - 45) students in Science.
4. This makes the number of Arts students twice the number of Science students: A - 20 = 2(S - 45).

We solve the equations:

1. From S = 2(A - 20), we get S = 2A - 40.
2. Substitute S in A - 20 = 2(S - 45):
3. A - 20 = 2((2A - 40) - 45)
4. Simplifying gives A - 20 = 4A - 170
5. Rearrange to solve for A: 3A = 150
6. A = 50 students initially in Arts.
7. From S = 2A - 40, we get S = 2(50) - 40 = 100 - 40 = 60 students initially in Science.

Now, calculate the number of students taking Economics:

1. In the remaining Arts, (A - 20) = 30 students are still taking Economics.
2. All 20 students who switched to MatEco take Economics.
3. All 45 students who switched to MatEco (from Science) take Economics.
4. Total number of students taking Economics: 30 + 20 + 45 = 95.

Therefore, the total number of students now taking Economics is 95.

Answer 2

Let the original number of Science students be S and Arts students be A.

Step 1: Analyze the first condition. When 20 students switch from Arts to MatEco, the remaining Arts students are A − 20. The total Science students become 2(A − 20), as the number of Science students is twice the number of Arts students.

S = 2(A − 20)

S = 2A − 40

Step 2: Analyze the second condition. When 45 Science students switch to MatEco, the remaining Science students are S − 45, and the number of Arts students becomes twice the number of Science students.

A − 20 = 2(S − 45)

Substituting S = 2A − 40 into the equation:

A − 20 = 2(2A − 40) − 45)

Simplify:

A − 20 = 2(2A − 85)

A − 20 = 4A − 170

3A = 150 =\(>\) A = 50

Step 3: Calculate S.

S = 2A − 40 = 2(50) − 40 = 60

Step 4: Calculate the number of students taking Economics. Students taking Economics are: - 20 students from Arts who switched to MatEco. - All remaining A − 20 Arts students. - 45 Science students who switched to MatEco.

Total: (A − 20) + 20 + 45 = A + 45 = 50 + 45 = 95

Answer: 95