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 (including blog)

We know 5 users visited news and shopping but not blog, and 8 users visited only news. The total for news site is 28. So users who visited news and shopping (including those who also visited blog) is \(28 - 8 - \text{only news and blog}\). Wait, first, let's find the number of users who visited news and shopping and blog. Wait, no, first, let's handle the news site.

Total news site visitors: 28. Only news: 8. News and shopping but not blog: 5. Let \(x\) be news and blog but not shopping, and \(y\) be all three (news, shopping, blog). So \(8 + 5 + x + y = 28\). So \(x + y = 28 - 8 - 5 = 15\).

Now shopping site: 31. News and shopping but not blog: 5. Shopping and blog but not news: 4. Let \(y\) be all three. So only shopping: \(31 - 5 - 4 - y\). Wait, only shopping would be \(31 - (5 + 4 + y)\). But we can also use the total users.

Total users: 49. Let \(z\) be only blog.

We have:

Only news: 8

News and shopping not blog: 5

Shopping and blog not news: 4

Only shopping: let's find it. Wait, shopping site: 31. So only shopping = 31 - (5 + 4 + y).

News site: 28 = 8 (only news) + 5 (news&shopping not blog) + x (news&blog not shopping) + y (all three). So \(x + y = 15\) as before.

Blog site: 26 = x (news&blog not shopping) + 4 (shopping&blog not news) + y (all three) + z (only blog). So \(x + y + 4 + z = 26\). But \(x + y = 15\), so \(15 + 4 + z = 26\)? Wait, no, wait: \(x + y + 4 + z = 26\) => \(z = 26 - 15 - 4 = 7\)? Wait, no, that can't be, because total users:

Total users = only news (8) + only shopping + only blog (z) + news&shopping not blog (5) + news&blog not shopping (x) + shopping&blog not news (4) + all three (y).

So total: \(8 + \text{only shopping} + z + 5 + x + 4 + y = 49\).

We know \(x + y = 15\), so substitute: \(8 + \text{only shopping} + z + 5 + 15 + 4 = 49\) => \(8 + 5 + 15 + 4 + \text{only shopping} + z = 49\) => \(32 + \text{only shopping} + z = 49\) => \(\text{only shopping} + z = 17\).

Now, shopping site: 31 = 5 (news&shopping not blog) + 4 (shopping&blog not news) + y (all three) + \(\text{only shopping}\). So \(5 + 4 + y + \text{only shopping} = 31\) => \(9 + y + \text{only shopping} = 31\) => \(y + \text{only shopping} = 22\).

From news site: \(x + y = 15\), and from blog site: \(x + y + 4 + z = 26\) => \(15 + 4 + z = 26\) => \(z = 7\)? Wait, that seems off. Wait, no, let's check total users again.

Wait, maybe a better approach with Venn diagrams.

Let’s define:

- \(N\) = news, \(S\) = shopping, \(B\) = blog.

Given:

- \(|N \cup S \cup B| = 49\)

- \(|N| = 28\), \(|S| = 31\), \(|B| = 26\)

- \(|N \cap S \cap eg B| = 5\)

- \(|S \cap B \cap eg N| = 4\)

- \(|N \cap eg S \cap eg B| = 8\)

We need to find \(|B \cap eg N \cap eg S| = z\) (only blog).

First, find \(|N \cap S \cap B| = y\) (all three), and \(|N \cap B \cap eg S| = x\) (news and blog, not shopping).

From \(|N| = 8 + 5 + x + y = 28\) => \(x + y = 28 - 8 - 5 = 15\) --- (1)

From \(|S| = |N \cap S \cap eg B| + |S \cap B \cap eg N| + |N \cap S \cap B| + |S \cap eg N \cap eg B|\)

So \(31 = 5 + 4 + y + |S \cap eg N \cap eg B|\)

Let \(a = |S \cap eg N \cap eg B|\) (only shopping), then \(31 = 5 + 4 + y + a\) => \(a = 22 - y\) --- (2)

From \(|N \cup S \cup B| = 8 + 5 + x + y + 4 + a + z = 49\)

Substitute \(x + y = 15\) (from 1) and \(a = 22 - y\) (from 2):

\(8 + 5 + 15 + 4 + (22 - y) + z = 49\)

Simplify: \(8 + 5 = 13; 13 + 15 = 28; 28 + 4 = 32; 32 + 22 = 54; 54 - y + z = 49\) => \(-y + z = -5\) => \(z = y - 5\) --- (3)

From \(|B| = x + y + 4 + z = 26\) (since \(|B| = |N \cap B \cap eg S| + |S \cap B \cap eg N| + |N \cap S \cap B| + |B \cap eg N \cap eg S|\))

Substitute \(x + y = 15\) (from 1):

\(15 + 4 + z = 26\) => \(19 + z = 26\) => \(z = 7\)

Wait, but from (3), \(z = y - 5\), so \(7 = y - 5\) => \(y = 12\). Let's check if this works.

From (1): \(x + y = 15\) => \(x = 15 - 12 = 3\)

From (2): \(a = 22 - y = 22 - 12 = 10\) (only shopping)

Now check total users:

Only news: 8

News&shopping not blog: 5

News&blog not shopping: 3

All three: 12

Shopping&blog not news: 4

Only shopping: 10

Only blog: 7

Sum: 8 + 5 + 3 + 12 + 4 + 10 + 7 = 8+5=13; 13+3=16; 16+12=28; 28+4=32; 32+10=42; 42+7=49. Perfect, matches total users.

Also check blog site: 3 (news&blog not shopping) + 4 (shopping&blog not news) + 12 (all three) + 7 (only blog) = 3+4=7; 7+12=19; 19+7=26. Correct.

News site: 8 (only news) + 5 (news&shopping not blog) + 3 (news&blog not shopping) + 12 (all three) = 8+5=13; 13+3=16; 16+12=28. Correct.

Shopping site: 5 (news&shopping not blog) + 4 (shopping&blog not news) + 12 (all three) + 10 (only shopping) = 5+4=9; 9+12=21; 21+10=31. Correct.

So the number of users who visited only the blog is 7.

Answer:

7