Each visitor to an amusement park needs to buy a ticket. Tickets can be Platinum, Gold, or Economy. Visitors are classified as Old, Middle-aged, or Young. The following facts are known about visitors and ticket sales on a particular day:

1. 140 tickets were sold.

2. The number of Middle-aged visitors was twice the number of Old visitors, while the number of Young visitors was twice the number of Middle-aged visitors.

3. Young visitors bought 38 of the 55 Economy tickets that were sold, and they bought half the total number of Platinum tickets that were sold.

4. The number of Gold tickets bought by Old visitors was equal to the number of Economy tickets bought by Old visitors.

**Question No. 1:**

If the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Platinum tickets, then which among the following could be the total number of Platinum tickets sold?

**Options:**

- 34
- 38
- 32
- 36

**Question No. 2:**

If the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Economy tickets, then the number of Old visitors buying Gold tickets was

**Question No. 3:**

If the number of Old visitors buying Gold tickets was strictly greater than the number of Young visitors buying Gold tickets, then the number of Middle-aged visitors buying Gold tickets was

**Question No. 4:**

Which of the following statements MUST be FALSE?

**Options:**

- The numbers of Old and Middle-aged visitors buying Economy tickets were equal
- The numbers of Old and Middle-aged visitors buying Platinum tickets were equal
- The numbers of Middle-aged and Young visitors buying Gold tickets were equal
- The numbers of Gold and Platinum tickets bought by Young visitors were equal

Number of young visitors = 2 x number of middle age visitors

Number of middle age visitors = 2 x number of old visitors

Total number of tickets sold = total number of visitors = 140

Hence, the number of young visitors = 80, the number of middle age visitors = 40 and the number of old visitors = 20

The given data can be tabulated as follows.

**Question 1**

Since half of the platinum tickets were purchased by young visitors, the remaining half were purchased by old and middle age visitors. Since these two are equal, half of total number of platinum tickets should be an even number. Among the given values, this is possible only for 32 and 36.

In case of 36, Old- Platinum = 9. In that case 2a = 11. But this is not possible. Hence, the total number of platinum tickets sold can only be 32.

Ans : 32

**Question 2**

Let Old – platinum = Middle aged – Economy = x

We get x + 2a = 20 and a + x + 38 = 55

By solving these two equations we get x = 3.

Ans : 3

**Question 3**

If the number of Old visitors buying Gold tickets was strictly greater than the number of Young visitors buying Gold tickets, then the number of Middle-aged visitors buying Gold tickets was

The maximum possible value of Young - gold = x – 1

Then young – platinum = 80 – (38 + x – 1) = 43 – x

Hence, Old – platinum + Middle age – Platinum = 43 – x

Total old + Middle age = 60

(Old – platinum + Middle age – platinum) + (Old – gold + Middle age – gold ) + (Old – economy + Middle age – economy) = 60

Hence, Old – gold + Middle age – gold = x

Thus, Middle age – gold = 0

Ans : Zero

**Question 4**

Since Old – Economy + Middle age – economy = 17, these two can never be equal. Hence, the statement that “The numbers of Old and Middle-aged visitors buying Economy tickets were equal” is false.

Ans : “The numbers of Old and Middle-aged visitors buying Economy tickets were equal”

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