Solution
Understanding the set :-
This is a logic puzzle involving categorisation and counting.
- We have two classifications for each visitor:
Ticket type — Platinum, Gold, Economy
Age group — Old, Middle-aged, Young
- We’re given:
a) Total tickets sold (140)
b) Relationship between age groups (Middle-aged are twice Old; Young are twice Middle-aged) — this fixes the ratio of visitors in each group.
c) Specific counts for certain ticket types for one group (e.g., Young bought 38 Economy, and half of all Platinum).
d) Equality condition for one group (Old bought equal numbers of Gold and Economy tickets).
- The goal in this kind of set is:
Use the given facts to restrict the possible values and ensure they fit the total visitors and ticket sales.
- The trick is:
a) Some facts are about totals (e.g., 55 Economy tickets sold overall).
b) Some facts are about proportions (e.g., half the Platinum tickets were bought by Young).
c) Some facts are equalities inside a row (e.g., Old’s Gold = Old’s Economy).
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | ||||
| Middle-aged | ||||
| Young | ||||
| Total | 140 |
Clue 1 says,
140 tickets were sold.
We can directly add this to the table.
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 | |||
| Middle-aged | 40 | |||
| Young | 80 | |||
| Total | 140 |
Clue 2 says,
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.
Let, the number of Old visitors be "x"
=> Middle aged visitors will be - "2x" (twice of old visitors)
Also, young visitors will be - "4x" (twice of Middle-aged visitors)
=> x + 2x + 4x =
=> x =
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 | |||
| Middle-aged | 40 | |||
| Young | 38 | 80 | ||
| Total | 55 | 140 |
Clue 3 says,
Young visitors bought 38 of the 55 Economy tickets that were sold
=> Total economy tickets = 55, out of which 38 were bought by Young.
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 | |||
| Middle-aged | 40 | |||
| Young | Y | 38 | 80 | |
| Total | 2Y | 55 | 140 |
Continuing with clue 3,
Young bought half the total number of Platinum tickets that were sold.
Let the total number of Platinum tickets sold be 2Y.
=> young bought half i.e., Y platinum tickets.
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 - 2Z | Z | Z | 20 |
| Middle-aged | 40 | |||
| Young | Y | 38 | 80 | |
| Total | 2Y | 55 | 140 |
Clue 4 says,
The number of Gold tickets bought by Old visitors was equal to the number of Economy tickets bought by Old visitors.
Now, Let the number of Gold tickets by old visitors be "Z".
=> Platinum tickets by old visitors =
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 - 2Z | Z | Z | 20 |
| Middle-aged | 40 | |||
| Young | Y | 42 - Y | 38 | 80 |
| Total | 2Y | 85 - 2Y | 55 | 140 |
Now, Young visitors buying gold tickets =
Also, Total number of Gold visitors =
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 - 2Z | Z | Z | 20 |
| Middle-aged | (Y + 2Z) - 20 | 43 - (Y+Z) | 17 - Z | 40 |
| Young | Y | 42 - Y | 38 | 80 |
| Total | 2Y | 85 - 2Y | 55 | 140 |
For Middle aged visitors :-
OLD - 2Y - Y - (20 - 2Z)
= (Y + 2Z) - 20
GOLD - 85 - 2Y - (Z - 42 + Y)
= 43 - (Y+Z)
ECONOMY - 55 - 38 - Z
= 17 - Z
| Platinum | Gold | Economy | Total | |
|---|---|---|---|---|
| Old | 20 - 2Z | Z | Z | 20 |
| Middle-aged | (Y + 2Z) - 20 | 43 - (Y+Z) | 17 - Z | 40 |
| Young | Y | 42 - Y | 38 | 80 |
| Total | 2Y | 85 - 2Y | 55 | 140 |
It is given that the number of Old visitors buying Gold tickets was strictly greater than the number of Young visitors buying Gold tickets.
Z > 42 - Y
⇒ Z + Y > 42 ... (1)
The number of Middle-aged visitors buying Gold tickets = 43 - (Y+Z)
Since (Y+Z) > 42, then We can say that (Y+Z) minimum = 43.
Hence, the number of Middle-aged visitors buying Gold tickets = 43 - 43 = 0