Example Case 4

Read the following and answer the questions that follow

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.
[CAT 2018 S2]

12) 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?

(1) 34
(2) 32
(3) 38
(4) 36

13) 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

Answer:

14) 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

Answer:

15) Which of the following statements MUST be FALSE?

(1) The numbers of Old and Middle-aged visitors buying Platinum tickets were equal
(2) The numbers of Middle-aged and Young visitors buying Gold tickets were equal
(3) The numbers of Gold and Platinum tickets bought by Young visitors were equal
(4) The numbers of Old and Middle-aged visitors buying Economy tickets were equal

Case Solution

*Statements form the case are provided in bold italics below.

Statement 1

140 tickets were sold.

Total of 140 tickets were sold in the park.

Statement 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..

Since the number of Middle-aged visitors was twice the number of Old visitors, let the number of Old visitors be $x$. Then the number of Middle-aged visitors will be $2x$. The number of Young visitors was twice the number of Middle-aged visitors, so the number of Young visitors will be $4x$.

Total Visitors = Old visitors + Middle-aged visitors + Young visitors = $x + 2x + 4x$ = 140
⇒ $x + 2x + 4x$ = 140
⇒ $7x$ = 140
⇒ $x$ = 20

So, The number of Old visitors, Middle-aged visitors and Young visitors are 20, 40 and 80 respectively.

Statement 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.

So, the total Economy tickets will be 55 and the Young visitors buying Platinum tickets will be 38.

Let us assume the total number of Platinum tickets as 2b, then the number of Platinum tickets bought by Young visitors will be b, since the Young visitors bought half the total number of Platinum tickets that were sold.

Statement 4

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

Let c be the number of Gold and Economy tickets bought by the Old visitors. Then the number of Platinum ticket bought by the Old visitors will be 20 – 2c.

From the above informations we can form a table,

We can find the remaining cells with the above table,

Number of Platinum tickets bought by Middle-aged visitors will be,

Total Platinum tickets – [Old Platinum tickets + Young Platinum tickets]
⇒ 2b – [(20 – 2c) + b]
⇒ b + 2c – 20

Number of Economy tickets bought by Middle-aged visitors will be,

Total Economy tickets – [Old Economy tickets + Young Economy tickets]
⇒ 55 – [c + 38]
⇒ 17 – c

Number of Gold tickets bought by Young visitors will be,

Total tickets bought by Young visitors – [Young Platinum tickets + Young Economy tickets]
⇒ 80 – [38 + b]
⇒ 42 – b

Number of Gold tickets bought by the visitors will be,

Total tickets – [Platinum tickets + Economy tickets]
⇒ 140 – [55+ 2b]
⇒ 85 – 2b
Number of Gold tickets bought by Middle-aged visitors will be,

Total Gold tickets – [Old Gold tickets + Young Gold tickets]
⇒ 85 – 2b – [c + (42 – b)]
⇒ 43 – b – c

Final table from the above informations,

12) Since the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Platinum tickets,

⇒ 20 – 2c = b + 2c – 20
⇒ b = 40 – 4c
⇒ b = 4(10 – c)

Since we have to find the total number of Platinum tickets which is 2b, we can multiply 2 on both sides

⇒ 2b = 8(10 – c)

Irrespective of the value of c, the value of 2b will be a 8 multiple. So the only 8 multiple in the option is 32

Answer: (2) 32 13) Since the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Economy tickets,

⇒ 20 – 2c = 17 – c
⇒ c = 3

The number of Old visitors buying Gold tickets (c) = 3

Answer: 3 14) The number of Old visitors buying Gold tickets was strictly greater than the number of Young visitors buying Gold tickets,

⇒ c > 42 – b
⇒ 42 – b – c > 0 --- (1)

Since we have to find the number of Middle-aged visitors buying Gold tickets which is 43 – b – c,

we can add +1 on both sides of (1)
⇒ 43 – b – c > 1
The maximum value can 43 – b – c take is 0. So the number of Middle-aged visitors buying Gold tickets will be 0.

Answer: 0

15)Let us consider the options.

Option (1)
The number of Old and Middle-aged visitors buying Platinum tickets were equal

⇒ 20 – 2c = b + 2c -20
⇒ 40 = b + 4c
Since there is no issue with this equation, we can’t say this is definitely FALSE.

Option (2)
The numbers of Middle-aged and Young visitors buying Gold tickets were equal

⇒ 43 – b – c = 42 – b
⇒ c = 1

Since c = 1 is possible, we can’t say this is definitely FALSE.

Option (3)
The numbers of Gold and Platinum tickets bought by Young visitors were equal

⇒ 42 – b = b
⇒ 2b = 42
⇒ b = 21

b = 21 is also a possible value, so we can’t say this will be definitely FALSE.

Option (4)
The numbers of Old and Middle-aged visitors buying Economy tickets were equal

⇒ c = 17 – c
⇒ 2c = 17
⇒ c = $\dfrac{17}{2}$= 8.5, which is not possible

So this will be definitely false.

Answer: (4) The numbers of Old and Middle-aged visitors buying Economy tickets were equal

Answer:
12) (2) 32
13) 3
14) 0
15) (4) The numbers of Old and Middle-aged visitors buying Economy tickets were equal