venn diagrams advandced math problem with solution | PDF - Free Printable

Let's solve this step by step using the information provided and fill in the Venn diagram accordingly.

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Given:

- Total people = 100
- 56 like Math
- 43 like Science
- 35 like Social Studies
- 18 like Math and Science
- 10 like Science and Social Studies
- 12 like Math and Social Studies
- 6 like all three subjects

We are to use this data to complete the Venn diagram and answer the questions.

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Step 1: Understand the overlaps

We’ll use the principle of inclusion-exclusion and work from the center outward.

Let:
- $ M $ = Math
- $ S $ = Science
- $ SS $ = Social Studies

We know:
- $ |M \cap S \cap SS| = 6 $

Now, we can find the only two-subject overlaps (excluding the triple overlap):

#### Math and Science only:
Total who like Math and Science = 18
But 6 of them also like Social Studies
So, Math and Science only = $ 18 - 6 = 12 $

#### Science and Social Studies only:
Total who like Science and Social Studies = 10
Subtract those who like all three: $ 10 - 6 = 4 $

#### Math and Social Studies only:
Total who like Math and Social Studies = 12
Subtract those who like all three: $ 12 - 6 = 6 $

Now, calculate the number of people who like only one subject:

#### Only Math:
Total who like Math = 56
Subtract:
- Math and Science only = 12
- Math and Social Studies only = 6
- All three = 6
So, Only Math = $ 56 - (12 + 6 + 6) = 56 - 24 = 32 $

#### Only Science:
Total who like Science = 43
Subtract:
- Math and Science only = 12
- Science and Social Studies only = 4
- All three = 6
So, Only Science = $ 43 - (12 + 4 + 6) = 43 - 22 = 21 $

#### Only Social Studies:
Total who like Social Studies = 35
Subtract:
- Science and Social Studies only = 4
- Math and Social Studies only = 6
- All three = 6
So, Only Social Studies = $ 35 - (4 + 6 + 6) = 35 - 16 = 19 $

Now let’s verify total number of people:

Add all regions:

- Only Math: 32
- Only Science: 21
- Only Social Studies: 19
- Math and Science only: 12
- Science and Social Studies only: 4
- Math and Social Studies only: 6
- All three: 6

Total = $ 32 + 21 + 19 + 12 + 4 + 6 + 6 = 100 $

✔ Matches total population.

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Now, answer the questions:

#### 6. How many people like Math only?
→ 32

#### 7. How many like the Science only?
→ 21

#### 8. If one person is chosen at random, what is the probability that that person will like Science and Math?
"Like Science and Math" means both, regardless of Social Studies.

This includes:
- Math and Science only: 12
- All three: 6
→ Total = $ 12 + 6 = 18 $

Probability = $ \frac{18}{100} = 0.18 $ or 18%

Answer: 0.18

#### 9. If one person is chosen at random, what is the probability that that person will like only Math?
Only Math = 32
Probability = $ \frac{32}{100} = 0.32 $

Answer: 0.32

#### 10. If one person is chosen at random, what is the probability that that person will not like Science?

First, find how many do not like Science.

Total = 100
Number who like Science = Only Science + Math & Science only + Science & Social only + All three
= $ 21 + 12 + 4 + 6 = 43 $

So, do not like Science = $ 100 - 43 = 57 $

Probability = $ \frac{57}{100} = 0.57 $

Answer: 0.57

#### 11. If one person is chosen at random, what is the probability that that person will like Science or Math?

"Or" means union: $ P(M \cup S) $

Use formula:
$ P(M \cup S) = P(M) + P(S) - P(M \cap S) $

We have:
- $ P(M) = \frac{56}{100} $
- $ P(S) = \frac{43}{100} $
- $ P(M \cap S) = \frac{18}{100} $

So,
$$
P(M \cup S) = \frac{56 + 43 - 18}{100} = \frac{81}{100} = 0.81
$$

Answer: 0.81

#### 12. If one person is chosen at random, what is the probability that that person will like Science but not math?

"Science but not Math" = Only Science + Science and Social Studies only
= $ 21 + 4 = 25 $

Probability = $ \frac{25}{100} = 0.25 $

Answer: 0.25

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✔ Final Answers:

6. 32
7. 21
8. 0.18
9. 0.32
10. 0.57
11. 0.81
12. 0.25

Let me know if you'd like a visual representation of the filled Venn diagram!