Calculate the number of schools subscribing to exactly two newspapers and only one newspaper using given subscription data | Step-by-Step Solution

Set Theory Problem: School Newspaper Subscriptions

What We're Solving:

We need to find how many schools subscribe to exactly two newspapers and how many subscribe to only one newspaper, given the total subscriptions and overlap information.

The Approach:

This is a classic Venn diagram problem using the principle of inclusion-exclusion to work backwards from what we know. Think of it like organizing overlapping circles - we need to figure out what's in each section without double-counting.

The key insight is that when we add up all individual subscriptions (69 + 62 + 55), we're counting some schools multiple times. Schools with 2 subscriptions get counted twice, and schools with all 3 get counted three times!

Step-by-Step Solution:

Step 1: Set up our variables

• Let A = schools with exactly 2 newspapers
• Let B = schools with exactly 1 newspaper
• We know: schools with exactly 3 newspapers = 32

Step 2: Use the total school count Since every school has at least one subscription: Total schools = (schools with 1) + (schools with 2) + (schools with 3) 87 = B + A + 32 Therefore: A + B = 55

Step 3: Count total subscriptions two ways When we add up individual newspaper subscriptions: 69 + 62 + 55 = 186 total subscriptions

But let's consider how many subscriptions each group contributes:

• Schools with 1 newspaper: B schools × 1 subscription each = B
• Schools with 2 newspapers: A schools × 2 subscriptions each = 2A
• Schools with 3 newspapers: 32 schools × 3 subscriptions each = 96

So: B + 2A + 96 = 186 Therefore: B + 2A = 90

Step 4: Solve the system of equations We now have:

• A + B = 55 (from Step 2)
• B + 2A = 90 (from Step 3)

Subtracting the first equation from the second: (B + 2A) - (A + B) = 90 - 55 A = 35

Substituting back: B = 55 - 35 = 20

The Answer:

• Schools with exactly two newspapers: 35
• Schools with only one newspaper: 20

Memory Tip:

When dealing with overlapping sets, remember the "subscription counting trick": if you add up all individual totals, you count each person once for each set they belong to. So people in 2 sets get counted twice, people in 3 sets get counted three times! This over-counting is actually useful information that helps us solve the problem.

Great job working through this! Set theory problems like this appear everywhere - from marketing surveys to medical studies. You're building skills that apply far beyond math class! 🌟