Calculate the number of schools with subscriptions to either exactly two newspapers or only one newspaper | Step-by-Step Solution

Hello! This is a fantastic Venn diagram problem that really helps you understand how overlapping sets work in real life. 📊

What We're Solving:

We need to find how many of the 87 schools subscribe to exactly two newspapers, and how many subscribe to only one newspaper, given the subscription data for Graphic, Mirror, and Times newspapers.

The Approach:

We'll use a systematic Venn diagram approach. Think of this like organizing overlapping circles - some schools appear in multiple circles (they subscribe to multiple papers), and we need to carefully count each section without double-counting anyone.

The key insight is: Total = Only one + Exactly two + All three

Step-by-Step Solution:

Step 1: Set up what we know

• Total schools: 87
• Graphic subscribers: 69
• Mirror subscribers: 62
• Times subscribers: 55
• All three papers: 32

Step 2: Find the sum of individual subscriptions 69 + 62 + 55 = 186 total subscriptions

Step 3: Understand why this sum is too high We have 186 subscriptions but only 87 schools! This means schools are being counted multiple times:

• Schools with exactly 2 papers are counted twice
• Schools with all 3 papers are counted three times

Step 4: Set up our equation Let's call:

• x = schools with exactly 2 papers
• y = schools with only 1 paper

We know: y + x + 32 = 87 (total schools)

Step 5: Account for the overcounting In our sum of 186:

• Schools with only 1 paper contribute: y × 1 = y
• Schools with exactly 2 papers contribute: x × 2 = 2x
• Schools with all 3 papers contribute: 32 × 3 = 96

So: y + 2x + 96 = 186

Step 6: Solve the system of equations From Step 4: y + x + 32 = 87, so y + x = 55 From Step 5: y + 2x + 96 = 186, so y + 2x = 90

Subtracting the first from the second: (y + 2x) - (y + x) = 90 - 55 x = 35

Therefore: y = 55 - 35 = 20

Step 7: Verify our answer

• Only one paper: 20 schools
• Exactly two papers: 35 schools
• All three papers: 32 schools
• Total: 20 + 35 + 32 = 87 ✓

The Answer:

a. 35 schools have subscriptions to exactly two newspapers b. 20 schools have subscriptions to only one newspaper

Memory Tip:

Remember the "overcounting principle"! When you add up individual set totals, you're counting overlaps multiple times. Think: "How many times does each type of school get counted in my sum?" This helps you set up the right equations every time! 🎯

Great job working through this complex problem - Venn diagram problems become much easier once you master this systematic approach!