Question

1. a total of 49 internet users visited at least 1 of 3 websites: a news site, a shopping site, and a blog. 28 users visited the news site. 31 users visited the shopping site. 26 users visited the blog. 5 users visited the news site and the shopping site but not the blog. 4 users visited the shopping site and the blog but not the news site. 8 users visited only the news site. how many users visited only the blog?

Explanation:

Step1: Find users who visited news and shopping and blog

First, find the number of users who visited news and shopping (including those who also visited blog). We know 8 visited only news, 5 visited news and shopping but not blog. So total news and shopping (including blog) is \(28 - 8 - 5=15\)? Wait, no. Wait, total news site visitors: 28. Only news: 8, news and shopping but not blog: 5. Let \(x\) be news, shopping, and blog. Then news site visitors: only news + news&shopping&not blog + news&blog&not shopping + news&shopping&blog = 28. So 8 + 5 + (news&blog&not shopping) + \(x\) = 28. But maybe better to use the principle of inclusion - exclusion.

Total users: 49. Let's define:

- Only news: \(N_{only}=8\)
- News and shopping only: \(N\cap S_{only}=5\)
- Shopping and blog only: \(S\cap B_{only}=4\)
- News: \(N = 28\), Shopping: \(S = 31\), Blog: \(B = 26\)
- Let \(N\cap S\cap B=x\) (all three)
- Let \(N\cap B_{only}=y\) (news and blog only)
- Let \(S\cap N\cap B=x\) (already defined)
- Let \(B_{only}=z\) (only blog, what we need to find)

First, for news site: \(N_{only}+N\cap S_{only}+N\cap B_{only}+N\cap S\cap B = N\)

So \(8 + 5 + y + x=28\) → \(y + x=15\) ...(1)

For shopping site: \(N\cap S_{only}+S\cap B_{only}+N\cap S\cap B+S_{only}=S\)

We need \(S_{only}\) (only shopping). Wait, total shopping: 31. So \(5 + 4 + x + S_{only}=31\) → \(S_{only}=31 - 5 - 4 - x=22 - x\) ...(2)

For blog site: \(N\cap B_{only}+S\cap B_{only}+N\cap S\cap B+B_{only}=B\)

So \(y + 4 + x + z=26\) → \(y + x + z=22\) ...(3)

Total users: \(N_{only}+N\cap S_{only}+S_{only}+S\cap B_{only}+N\cap B_{only}+N\cap S\cap B+B_{only}=49\)

Substitute the knowns:

\(8 + 5 + (22 - x) + 4 + y + x + z=49\)

Simplify: \(8 + 5 + 22 + 4 + y + z=49\) (since \(-x + x=0\))

\(39 + y + z=49\) → \(y + z=10\) ...(4)

From equation (1): \(y + x=15\) → \(x = 15 - y\)

From equation (3): \(y + x + z=22\). Substitute \(x = 15 - y\):

\(y + (15 - y) + z=22\) → \(15 + z=22\) → \(z=7\)? Wait, no, that can't be. Wait, maybe I made a mistake in total users breakdown.

Wait, the total users who visited at least one site: only N + only S + only B + N∩S only + N∩B only + S∩B only + N∩S∩B = 49.

Let's list all regions:

1. Only N: 8
2. N∩S only: 5
3. Only S: let's call it S_only
4. S∩B only: 4
5. Only B: z (what we need)
6. N∩B only: y
7. N∩S∩B: x

Now, for N (news site): 8 (only N) + 5 (N∩S only) + y (N∩B only) + x (all three) = 28 → 13 + y + x = 28 → y + x = 15 ...(A)

For S (shopping site): 5 (N∩S only) + S_only (only S) + 4 (S∩B only) + x (all three) = 31 → 9 + S_only + x = 31 → S_only = 22 - x ...(B)

For B (blog site): y (N∩B only) + 4 (S∩B only) + z (only B) + x (all three) = 26 → y + 4 + z + x = 26 → y + z + x = 22 ...(C)

Total users: 8 + 5 + S_only + 4 + y + x + z = 49 → 17 + S_only + y + x + z = 49 → S_only + y + x + z = 32 ...(D)

Substitute S_only from (B) into (D): (22 - x) + y + x + z = 32 → 22 + y + z = 32 → y + z = 10 ...(E)

From (A): y + x = 15 → x = 15 - y

Substitute x into (C): y + z + (15 - y) = 22 → z + 15 = 22 → z = 7? Wait, but let's check with blog total. If z=7, y + x=15, and y + z=10 (from E), then y=10 - z=10 -7=3. Then x=15 - y=12. Then blog site: y + 4 + z + x=3 + 4 +7 +12=26. Correct. Shopping site: 5 + S_only +4 +12=31 → S_only=31 -5 -4 -12=10. Then total users: 8 +5 +10 +4 +3 +12 +7=49. 8+5=13, 13+10=23, 23+4=27, 27+3=30, 30+12=42, 42+7=49. Correct. So only blog is 7? Wait, but let's re - check.

Wait, another way:

Total news site: 28. Only news:8, news&shopping&not blog:5. So news&blog (including all three): 28 -8 -5=15. So news&blog (only news&blog + all three)=15.

Shopping site:31. News&shopping&not blog:5, shopping&blog&not news:4. So shopping&news (including all three) + shopping&only: 31 -5 -4=22. So shopping&only + (news&shopping&all three)=22.

Blog site:26. Shopping&blog&not news:4, so blog&news (only news&blog + all three) + blog&only=26 -4=22. So (news&blog&only + all three) + blog&only=22. But news&blog&only + all three=15 (from news site). So 15 + blog&only=22 → blog&only=7. Yes, that's simpler.

So:

From news site: only N (8) + N&S only (5) + (N&B only + N&S&B) =28 → (N&B only + N&S&B)=28 -8 -5=15.

From blog site: (N&B only + N&S&B) + S&B only (4) + only B=26 → 15 +4 + only B=26 → only B=26 -19=7.

Yes, that's correct.

Step2: Verify with total users

Total users: only N (8) + N&S only (5) + only S + S&B only (4) + N&B only + N&S&B + only B=49.

We know only N=8, N&S only=5, S&B only=4, only B=7, (N&B only + N&S&B)=15.

So 8 +5 + only S +4 +15 +7=49 → only S +39=49 → only S=10.

Now check shopping site: only S (10) + N&S only (5) + S&B only (4) + N&S&B=10 +5 +4 + (15 - N&B only). Wait, no, N&S&B is part of (N&B only + N&S&B)=15. Let N&B only = a, N&S&B = b. Then a + b=15.

Shopping site: only S (10) + N&S only (5) + S&B only (4) + b=31 → 10 +5 +4 +b=31 → b=12. Then a=15 -12=3.

Now check all regions:

- Only N:8
- N&S only:5
- Only S:10
- S&B only:4
- N&B only:3
- N&S&B:12
- Only B:7

Sum:8+5=13, +10=23, +4=27, +3=30, +12=42, +7=49. Correct.

Answer:

7