Question

In a university club of 200 people, the number of Political Science majors is 50 less than 4 times the number of International Relations majors. If one fifth of the club members are neither Political Science majors nor International Relations majors, and no club member is majoring in both Political Science and International Relations, how many of the club members are International Relations majors?

• A. 42
• B. 50
• C. 71
• D. 95
• E. 124

Hint:

When faced with word problems, the first step should always be to translate the sentences into mathematical equations. Clearly define your variables to avoid confusion. Breaking down the problem into smaller, manageable pieces makes it easier to solve.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a word problem that can be solved using a system of linear equations derived from the given information. It involves concepts from set theory, specifically dealing with disjoint sets (since no member majors in both).
Step 2: Key Formula or Approach:
Let's define our variables:
- Let \(I\) be the number of International Relations majors.
- Let \(P\) be the number of Political Science majors.
- Let \(T\) be the total number of people in the club.
- Let \(N\) be the number of people majoring in neither subject.
The total number of people is the sum of those in each distinct group: \( T = P + I + N \).
Step 3: Detailed Explanation:
From the problem statement, we can extract the following information:
1. Total members, \( T = 200 \).
2. The number of Political Science majors in terms of International Relations majors: \( P = 4I - 50 \).
3. The number of members majoring in neither subject is one-fifth of the total: \( N = \frac{1}{5} \times 200 = 40 \).
4. No club member majors in both, so the sets P and I are disjoint.
First, let's find the total number of members who are majoring in either Political Science or International Relations.
\[ \text{Total Majors} = T - N = 200 - 40 = 160 \] Since no one majors in both, the total number of majors is simply the sum of the individual majors:
\[ P + I = 160 \] Now we have a system of two linear equations with two variables:
Equation (i): \( P + I = 160 \)
Equation (ii): \( P = 4I - 50 \)
We can solve this system by substituting Equation (ii) into Equation (i):
\[ (4I - 50) + I = 160 \] Combine the terms with \(I\):
\[ 5I - 50 = 160 \] Add 50 to both sides:
\[ 5I = 160 + 50 \] \[ 5I = 210 \] Divide by 5 to solve for \(I\):
\[ I = \frac{210}{5} = 42 \] Step 4: Final Answer:
The number of club members who are International Relations majors is 42.