# PE 3 - Venn Diagram | LR - Venn Diagram

**Answer the next 5 questions based on the information given below.**

Out of a group of 490 toursits, 210 visited London, 190 visited Hong Kong and 190 visited Singapore. 30 of them visited all three cities, while 380 visited exactly one of the three cities. The number of toursits who visited exactly two out of the three cities is three times as many as those who did not visit any one of the three cities.

**PE 3 - Venn Diagram | LR - Venn Diagram**

How many toursits have not visited any one of the three cities?

- (a)
40

- (b)
20

- (c)
30

- (d)
50

- (e)
None of these

Answer: Option B

**Explanation** :

From the given data, we get the following diagram.

Let a, b, c denote the number of people who visited exactly one city; x, y, z denote the number of people who visited exactly two cities and w denote the number of people who did not visit any of the three cities.

∴ x + y + z = 3w ... (1)

a + b + c = 380 .. (2)

(a + b + c) + (x + y + z) + 30 + w = 490

∴ 380 + 3w + 30 + w = 490

∴ 4w + 410 = 490

∴ 4w = 80 i.e. w = 20

Hence, the number of toursits who have not visited any one of the three cities is 20.

Hence, option (b).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

How many toursits visited not more than one city?

- (a)
100

- (b)
200

- (c)
300

- (d)
400

- (e)
None of these

Answer: Option D

**Explanation** :

Consider the solution to first question of this set.

The number of toursits who did not visit more than one city = (Number of people who did not visit any city) + (Number of people who visited exactly one city)

= 20 + (a + b + c)

= 20 + 380 = 400

Hence, option (d).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

If the number of toursits who have visited at least one of the two cities Hong Kong and Singapore is 330, how many toursits visited only Hong Kong and Singapore?

- (a)
40

- (b)
60

- (c)
20

- (d)
30

- (e)
50

Answer: Option C

**Explanation** :

Consider the solution to first question of this set.

The number of toursits who visited only Hong Kong and Singapore is denoted by z in the Venn diagram drawn above.

The set of toursits who visited at least one city from Hong Kong and Singapore implies the union of the two sets H and S.

This set is given as: S ∪ H = S + H − (S ∩ H)

∴ 330 = 190 + 190 − (z + 30)

∴ z = 380 − 30 – 330 = 20

Hence, 20 people visited only Hong Kong and Singapore

Hence, option (c).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

If 360 toursits visited at least one of the two cities Hong Kong and London, how many toursits visited only Singapore?

- (a)
110

- (b)
80

- (c)
70

- (d)
120

- (e)
100

Answer: Option A

**Explanation** :

Consider the solution to first question of this set.

The number of toursits who visited only Singapore is denoted by c.

Since, 360 toursits visited at least one of the two cities Hong Kong and London,

∴ H ∪ L = H + L − (H ∩ L)

∴ 360 = 190 + 210 − (x + 30)

∴ x = 10.

Now, it is already found that x + y + z = 3w = 60.

∴ y + z = 60 − 10 = 50.

Now, S = c + (y + z) + 30 = 190

∴ c + 50 + 30 = 190 i.e. c = 110.

Hence, the number of toursits who visited only Singapore = 110.

Hence, option (a).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

If there is nobody who visited only London and Singapore, how many people visited only Hong Kong?

- (a)
160

- (b)
180

- (c)
140

- (d)
120

- (e)
100

Answer: Option E

**Explanation** :

Consider the solution to first question of this set.

x + y + z = 60.

Since no pilgrim visited only London and Singapore, y = 0

∴ x + z = 60.

Also, the number of toursits who visited only Hong Kong is denoted by b.

b + (x + z) + 30 = 190

∴ b + 60 + 30 = 190

∴ b = 100

Hence, 100 people visited only Hong Kong

Hence, option (e).

Workspace:

**Answer the next 4 questions based on the information given below.**

In August, 740 tourists visited Madhya Pradesh. Each tourist went to at least one of the three places – Panchmari, Khajuraho or hanumantia.

Following information is known about these tourists.

- The number of tourists visiting Panchmari alone is equal to the tourists who also visited Khajuraho.
- The number of tourists visiting Khajuraho alone is double the number of tourists visiting all the three places.
- 340 tourists visited Hanumantia.
- The number of tourists visiting Hanumantia alone is twenty less than the number of tourists visiting Khajuraho alone.
- 200 tourists who visited Hanumantia also visited at least one more place.
- Of the three places, maximum number of tourists went to Panchmari.

**PE 3 - Venn Diagram | LR - Venn Diagram**

Based on the information given above, the minimum number of tourists visiting both Panchmari and Hanumantia, but not Khajuraho is:

Answer: 61

**Explanation** :

Let the number of people visiting Panchmari alone = x.

From (1): Number of tourists visiting Panchmari and Khajuraho is also x.

Let the number of people visiting all three places = a.

∴ Number of people visiting only Panchmari and Khajuraho = x – a.

From (2): Number of people visiting Khajuraho alone = 2a.

From (4): Number of people visiting Hanumantia alone = 2a – 20.

From (3 and 5): Number of people visiting Hanumantia alone = 340 – 200 = 140.

⇒ 2a – 20 = 140

⇒ a = 80

Total number of people visiting Madhya Pradesh = 740 = 340 + x + (x – a) + 2a

⇒ 740 = 340 + 2x + a

⇒ 2x + 80 = 400

⇒ x = 160

From (5): b + a + c = 200

⇒ b + c = 120 …Eq. (1)

Now, number of people visiting

Panchmari = 2x + b = 320 + b, and

Khajuraho = 2a + x + c = 320 + c

From (6): 320 + b > 320 + c

⇒ b > c

∴ Minimum value of b is 61

∴ Minimum number of tourists visiting both Panchmari and Hanumantia, but not Khajuraho = b = 61.

Hence, 61.

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

Which of the following additional information would enable to find the exact number of tourists visiting various places?

- (a)
400 tourists are visiting Panchmari

- (b)
80 tourists are visiting all the three places

- (c)
460 tourists are visiting exactly one

- (d)
No need for any additional information

Answer: Option A

**Explanation** :

Consider the solution to first question of this set.

We don’t know the value of b and c so far. Hence, if 400 students visit Panchmari (option (a)) we can determine the values of b and c as well.

Hence, option (a).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

Next year these tourists decided to visit the same places as they did last year with the following exceptions. The tourists who visited all the three places visited only two of the three places. As a result, 20 tourists opted out of the Hanumantia, 20 opted out of the Khajuraho, while the remaining ones visiting all the three places opted out of the Panchmari. Which of the following statements, then, necessarily follows?

- (a)
The lowest number of tourists is now in Hanumantia

- (b)
More tourists are now in Panchmari as compared to Khajuraho

- (c)
More tourists are now in Hanumantia as compared to Khajuraho

- (d)
None of the above

Answer: Option D

**Explanation** :

Consider the solution to first question of this set.

Of total 80 people visiting all the three places:

20 opted out of Hanumantia, hence they visited only Panchmari and Khajuraho

20 opted out of Khajuraho, hence they visited only Panchmari and Hanumantia

40 opted out of Panchmari, hence they visited only Khajuraho and Hanumantia

We can draw the following Venn diagram now,

Number of tourists in

Panchmari = 280 + b > 340 [∵ b > 60]

Khajuraho = 300 + c < 360 [∵ c < 60]

Hanumantia = 200 + b + c = 320

Option (a): If c = 0, then this statement will not be true.

Option (b): If b = 61 and c = 60, then this statement will not be true.

Option (c): If c = 0, then this statement will not be true.

Hence, option (d).

Workspace:

**PE 3 - Venn Diagram | LR - Venn Diagram**

Next year these tourists decided to visit the same places as they did last year with the following exceptions. The tourists who visited all the three places visited only two of the three places. As a result, 20 tourists opted out of the Hanumantia, 20 opted out of the Khajuraho, while the remaining ones visiting all the three places opted out of the Panchmari.

Apart from the old tourists, some new tourists also visited Madhya Pradesh next year in August. Each of the new tourist visited only one place such that the number of tourists visiting each of the three places alone became identical. At this point, it was also found that the number of tourists visiting Panchmari and Khajuraho was the same as the number of tourists visiting Panchmari and Hanumantia. Which of the places now has the highest number of tourists?

- (a)
Khajuraho

- (b)
Panchmari

- (c)
Hanumantia

- (d)
Cannot be determined

Answer: Option B

**Explanation** :

Consider the solution to previuos question of this set.

Since the number of tourists visiting Panchmari and Khajuraho was the same as the number of tourists visiting Panchmari and Hanumantia, it means b = 80 and c = 40.

Number of tourists visiting

Panchmari = S + 200

Khajuraho = S + 180

Hanumantia = S + 180

Hence, option (b).

Workspace:

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