Using Venn Diagrams Problems Independent Practice Worksheet 1 ... - Free Printable

Let’s solve each problem one by one. We’ll use Venn diagrams in our mind (or on paper) to help us visualize the groups and overlaps.

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Problem 1:

There are 49 students total.
25 attend Woodshop class.
20 attend Word Processing class.
7 are in both classes.

We want to find how many are not enrolled in either.

First, add the number in Woodshop and Word Processing:
25 + 20 = 45

But this counts the 7 students who are in both classes twice, so we subtract them once:
45 - 7 = 38 → These are the students enrolled in at least one class.

Now subtract from total students:
49 - 38 = 11

✔ So, 11 students are not enrolled in either class.

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Problem 2:

Group of 20 friends.
7 were playing basketball.
9 were playing tennis.
5 were playing nothing.

We want to know how many were playing basketball only.

First, find how many were playing something:
20 total - 5 doing nothing = 15 playing something.

Of those 15, some played both sports. Let’s call the number who played both “x”.

Then:
Basketball only + Tennis only + Both = 15
(7 - x) + (9 - x) + x = 15
→ 7 - x + 9 - x + x = 15
→ 16 - x = 15
→ x = 1

So, 1 person played both.

Therefore, basketball only = 7 - 1 = 6

✔ Answer: 6 friends were playing basketball only.

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Problem 3:

Total people: 90
About to buy Nike shoes: 18
About to buy Reebok shoes: 12
About to buy both brands: 10

We want to know how many were buying neither.

Use the formula:
Nike only + Reebok only + Both + Neither = Total

Nike only = 18 - 10 = 8
Reebok only = 12 - 10 = 2
Both = 10

So, people buying at least one brand: 8 + 2 + 10 = 20

Neither = 90 - 20 = 70

✔ Answer: 70 people were buying neither brand.

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Problem 4:

Total students: 402
Engineering majors: 50% → 0.5 × 402 = 201
Dogs: 25% → 0.25 × 402 = 100.5 → Wait! That can’t be right — you can’t have half a student.

Hold on — let’s check: 25% of 402 is 100.5? That suggests maybe the numbers are meant to be exact. Maybe it’s okay for now — perhaps we’re supposed to round? But actually, let’s re-read:

It says “how many students are majoring in both fields?” — meaning engineering AND dogs? That doesn’t make sense — “fields” probably means engineering and... wait, no — rereading:

“50% of students are majoring in engineering and 25% of students are majoring in dogs.” — that must be a typo or misphrase. Probably meant “owning dogs” or “studying dog-related field”? But then it says “majoring in both fields” — implying two academic fields.

Wait — looking again: “majoring in engineering and 25% of students are majoring in dogs” — that seems odd. Perhaps it’s “majoring in engineering and 25% are owning dogs”? But then “both fields” wouldn’t fit.

Actually, let’s assume it’s a mistake and it should be:

“50% are majoring in engineering, 25% are majoring in biology (or another field), and 10% are majoring in both.”

But the problem says: “50% of students are majoring in engineering and 25% of students are majoring in dogs. 10% of students are majoring in both fields.”

This is confusing. “Majoring in dogs” isn’t a real major. Likely, it’s a typo, and it should be “owning dogs” or “studying veterinary science” or something. But since it says “both fields”, I think we have to treat “dogs” as a second major — even if unrealistic.

So, let’s proceed with the math as given:

Total = 402
Engineering = 50% → 201
Dogs = 25% → 100.5 → Hmm. This is problematic.

Wait — 25% of 402 is 100.5? That’s not possible. Maybe the percentages are approximate? Or maybe it’s 25% of the whole, but we need to use whole numbers.

Alternatively, perhaps the 10% is of the total, and we can compute:

Let E = engineering = 50% of 402 = 201
D = dogs = 25% of 402 = 100.5 → Not integer.

This suggests there might be an error in the problem. But let’s try using fractions:

50% = 1/2 → 402 / 2 = 201
25% = 1/4 → 402 / 4 = 100.5 → Still not integer.

Perhaps the problem meant 25% of the engineering students? No, it says “25% of students”.

Another possibility: maybe “dogs” is not a major, but pet ownership, and “both fields” is a misnomer. But the question asks “how many students are majoring in both fields?” — so it implies two majors.

Given the inconsistency, let’s assume the numbers are meant to be used as is, and perhaps we round.

But 10% of 402 = 40.2 → also not integer.

This is messy. Let’s calculate exactly:

E = 0.5 * 402 = 201
D = 0.25 * 402 = 100.5
Both = 0.10 * 402 = 40.2

But you can’t have 0.5 or 0.2 students. So likely, the problem has a typo. Perhaps total students is 400? Let’s check with 400:

If total = 400:
E = 200
D = 100
Both = 40
Then both fields = 40.

But the problem says 402. Maybe it’s 402, and we’re to report decimal? Unlikely.

Perhaps “25% of students are majoring in dogs” is wrong — maybe it’s “25% of engineering students”? But the text doesn’t say that.

Looking back at the original image text: “50% of students are majoring in engineering and 25% of students are majoring in dogs. 10% of students are majoring in both fields.”

I think we have to go with the math as written, even if unrealistic.

So, number majoring in both = 10% of 402 = 40.2 → which is not possible.

Wait — perhaps “10% of students are majoring in both” — so that’s given directly.

The question is: “How many students are majoring in both fields?”

And it says “10% of students are majoring in both fields.”

So, regardless of the other numbers, if 10% are in both, then answer is 10% of 402.

10% of 402 = 40.2 → still not integer.

This is a problem. Perhaps it’s 400 students? Or maybe we’re to round to nearest whole number.

In real homework, sometimes they expect you to use the percentage directly.

10% of 402 = 40.2 → but since students are whole, maybe 40.

But let’s see the context. In problem 1, numbers worked out evenly. Here, not.

Another thought: perhaps “25% of students are majoring in dogs” is a distractor, and we only need the 10% for both.

The question is: “How many students are majoring in both fields?”

And it explicitly says “10% of students are majoring in both fields.”

So, answer should be 10% of 402.

Calculate: 0.10 * 402 = 40.2

Since we can't have 0.2 student, and this is likely a typo, but for the sake of answering, I'll assume it's 40.

Or perhaps the total is 400? Let me check if 402 is correct.

Looking at the image text: "There are 402 students in an institute."

Yes, 402.

Perhaps in such cases, we report the exact value, but that doesn't make sense for students.

Maybe "10%" is of the total, and we calculate 40.2, but since it's impossible, the problem might have intended 400.

To resolve, let's notice that in problem 5 and 6, numbers are small and work out.

For this, I think the intended answer is 10% of 402 = 40.2, but since it's students, perhaps 40.

But let's do exact calculation: 10/100 * 402 = 4020/100 = 40.2

I think there's a mistake in the problem, but for now, I'll go with 40 as the closest whole number.

However, let's read the question again: "How many students are majoring in both fields?"

And it says "10% of students are majoring in both fields." So if we take that literally, it's 40.2, but that's not possible.

Perhaps "both fields" refers to engineering and dogs, and the 10% is given, so we use that.

I think for the purpose of this exercise, we'll calculate 10% of 402 = 40.2, but since it's a count, maybe it's 40.

But let's move on and come back.

Actually, upon second thought, in Venn diagram problems, sometimes the "both" is given, and we don't need the individual percentages for the question.

The question is only asking for the number in both, which is given as 10% of total.

So, 10% of 402 = 40.2

But since students are whole, and 402 is even, 10% should be 40.2, which is not integer, so likely a typo.

Perhaps it's 400 students. Let me assume that for now, as 400 makes sense.

If total = 400, then 10% = 40.

I think that's what was intended.

So, I'll go with 40.

✔ Answer: 40 students are majoring in both fields. (assuming total is 400 or rounding)

But to be precise, let's calculate with 402:

10% of 402 = 40.2 — not possible, so perhaps the answer is 40.

I'll box 40 for now.

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Problem 5:

10 students taking Mark’s class.
7 students taking Jackson’s class.
12 students taking both classes.

Wait — that can't be. If 12 are taking both, but only 10 are in Mark’s class, that's impossible because you can't have more in both than in one class.

Let’s read: "10 students are taking Mark’s class, 7 students are taking Jackson’s class, 12 students are taking both classes."

That’s logically impossible. The number in both cannot exceed the number in either single class.

So, likely a typo. Probably, it's 10 in Mark’s, 7 in Jackson’s, and some number in both, but it says 12 in both.

Perhaps it's 10 in Mark’s only, 7 in Jackson’s only, and 12 in both? But the text doesn't say "only".

The text says: "10 students are taking Mark’s class" — this usually means total in Mark’s, including those in both.

Similarly for Jackson’s.

So, if |M| = 10, |J| = 7, |M ∩ J| = 12, but 12 > 10 and 12 > 7, impossible.

So, definitely a typo.

Perhaps it's 10 in Mark’s, 7 in Jackson’s, and 2 in both? Or something.

Another possibility: "12 students are taking both" is wrong; maybe it's "2 students are taking both".

Or perhaps the numbers are switched.

Let’s look at the question: "How many total students are there in total between the two classes?"

If we assume that "10 students are taking Mark’s class" means only Mark’s, and "7 students are taking Jackson’s class" means only Jackson’s, and "12 students are taking both", then total = 10 + 7 + 12 = 29.

But typically, when it says "taking Mark’s class", it includes those taking both, unless specified "only".

In standard interpretation, if it says "10 students are taking Mark’s class", it means the total in Mark’s class is 10, which includes those also in Jackson’s.

But here, if both is 12, that's larger than 10, impossible.

So, likely, the "12" is a typo, and it should be a smaller number.

Perhaps it's 10 in Mark’s, 7 in Jackson’s, and x in both, and we need to find total, but it gives both as 12, which is inconsistent.

Another idea: perhaps "12 students are taking both" is correct, and the 10 and 7 are for only one class.

Let me check the wording: "10 students are taking Mark’s class, 7 students are taking Jackson’s class, 12 students are taking both classes."

This is ambiguous. In some contexts, "taking Mark’s class" might mean exclusively, but usually not.

To resolve, let's assume that the 10 and 7 include the both group, but then both can't be 12.

Perhaps it's 10 in Mark’s only, 7 in Jackson’s only, and 12 in both — then total = 10+7+12=29.

And the question is "how many total students are there in total between the two classes?" which would be the union, so 29.

And that makes sense numerically.

Whereas if 10 is total in Mark’s, then both can't be 12.

So, I think the intended meaning is:

- 10 students are taking only Mark’s class
- 7 students are taking only Jackson’s class
- 12 students are taking both classes

Then total students = 10 + 7 + 12 = 29

✔ Answer: 29 students.

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Problem 6:

Out of 18 people in the Oke class.
11 have dogs.
7 have cats.
Four people have both cats and dogs.

We want to know how many club members have only cats.

First, total people = 18

Have dogs = 11
Have cats = 7
Have both = 4

So, have only cats = total cats - both = 7 - 4 = 3

We can verify:
Only dogs = 11 - 4 = 7
Only cats = 7 - 4 = 3
Both = 4
Total with pets = 7 + 3 + 4 = 14
So, no pets = 18 - 14 = 4, which is fine.

The question is only for "only cats", which is 3.

✔ Answer: 3

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Now, summarizing all answers:

1. 11
2. 6
3. 70
4. 40 (assuming 10% of 402 rounded or typo)
5. 29 (assuming 10 only Mark’s, 7 only Jackson’s, 12 both)
6. 3

For problem 4, to be accurate, let's calculate 10% of 402 = 40.2, but since it's students, and likely a typo, I'll use 40.

Perhaps the problem meant 10% of the engineering students or something, but as written, it's 10% of total.

Another way: perhaps "10% of students are majoring in both fields" is the key, so answer is 40.2, but we report 40.

I think for consistency, we'll go with 40.

Final Answers:

1. 11
2. 6
3. 70
4. 40
5. 29
6. 3

Now, writing the final answer section.

Note: For problem 4 and 5, there are inconsistencies, but based on common interpretations, these are the best answers.

Final Answer:
1. 11
2. 6
3. 70
4. 40
5. 29
6. 3