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CAT 2018 Slot 2 DILR · Set 7 of 8Logical ReasoningModerate DifficultyQ25–Q28 · 2 MCQ + 2 TITA

CAT 2018 Slot 2 DILR Set 7: Math Puzzles (4 Questions) with Solutions

The Math Puzzles set from the official CAT 2018 Slot 2 DILR paper: the caselet, all 4 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2018 Slot 2 DILR · Questions 25–28
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
140140 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 3838 of the 5555 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.

In the exam this set competed with 7 others for 60 minutes. Budget about 8 minutes if you attempt it.

Q25. Math Puzzles

MCQ · +3 / −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?
  1. A) 34
  2. B) 38
  3. C) 36
  4. D) 32
Correct Answer
D) 32
Step-by-step solution
📌 Core Concept
Set up the age groups from the ratio fact, express every ticket count in terms of one unknown (the number of Old visitors with Economy tickets), then apply the extra condition.
🔢 Step-by-Step Solution
1
Let Old =x=x, Middle-aged =2x=2x, Young =4x=4x. Then 7x=1407x=140, so Old =20=20, Middle-aged =40=40, Young =80=80.
2
Young bought 38 of the 55 Economy tickets, so Old and Middle-aged together bought 55−38=1755-38=17 Economy tickets.
3
Let Old visitors buy ee Economy tickets. Clue 4 gives ee Gold tickets as well, so Old Platinum =20−2e=20-2e.
4
Let PP be the total Platinum tickets. Young bought P/2P/2 of them, so Old Platinum + Middle-aged Platinum =P/2=P/2.
5
The question's condition makes Old Platinum equal to Middle-aged Platinum, so 2(20−2e)=P/22(20-2e)=P/2. This gives P=80−8eP=80-8e.
6
PP must be 80−8e80-8e for a whole number ee, so PP is a multiple of 8. Among the options only 32 fits, with e=6e=6. The values 34, 36 and 38 would need ee to be 5.755.75, 5.55.5 and 5.255.25.
Quick Check
With e=6e=6: Old have 8 Platinum, 6 Gold, 6 Economy (total 20). Middle-aged have 8 Platinum, 11 Economy and 21 Gold (total 40). Young have 16 Platinum, 38 Economy and 26 Gold (total 80). Economy totals 55 and all counts are non-negative.
Correct Answer: D

Q26. Math Puzzles

TITA · +3 / 0
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
Correct Answer
3
Step-by-step solution
📌 Core Concept
Fix the group sizes, express each group's ticket counts through the number of Old visitors with Economy tickets, then use the given equality to solve for it.
🔢 Step-by-Step Solution
1
Old =x=x, Middle-aged =2x=2x, Young =4x=4x, and 7x=1407x=140. So Old =20=20, Middle-aged =40=40, Young =80=80.
2
Young bought 38 of the 55 Economy tickets, so Old and Middle-aged together bought 17 Economy tickets.
3
Let Old visitors buy ee Economy tickets. Clue 4 gives them ee Gold tickets too, so Old Platinum =20−2e=20-2e.
4
Middle-aged Economy =17−e=17-e.
5
The question's condition says Old Platinum equals Middle-aged Economy: 20−2e=17−e20-2e=17-e, so e=3e=3.
6
Old visitors buying Gold tickets =e=3=e=3.
Quick Check
Old then have 14 Platinum, 3 Gold and 3 Economy, which sum to 20. Middle-aged Economy is 17−3=1417-3=14, which equals Old Platinum.
Correct Answer: 3

Q27. Math Puzzles

TITA · +3 / 0
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
Correct Answer
0
Step-by-step solution
📌 Core Concept
Write the Middle-aged Gold count and the Young Gold count in terms of the same two unknowns. The strict inequality then squeezes Middle-aged Gold to zero.
🔢 Step-by-Step Solution
1
Old =20=20, Middle-aged =40=40, Young =80=80 (from 7x=1407x=140).
2
Old and Middle-aged together bought 55−38=1755-38=17 Economy tickets. Let Old Economy =e=e, so Old Gold =e=e, Old Platinum =20−2e=20-2e and Middle-aged Economy =17−e=17-e.
3
Let PP be the total Platinum tickets. Young Platinum =P/2=P/2, so Old Platinum + Middle-aged Platinum =P/2=P/2. Hence Middle-aged Platinum =P/2−(20−2e)=P/2-(20-2e).
4
Young Gold =80−38−P/2=42−P/2=80-38-P/2=42-P/2.
5
Middle-aged Gold =40−(17−e)−(P/2−20+2e)=43−e−P/2=40-(17-e)-\left(P/2-20+2e\right)=43-e-P/2.
6
Compare the two: Middle-aged Gold =1+(Young Gold)−e=1+\text{(Young Gold)}-e.
7
The condition says Old Gold is strictly more than Young Gold, so e≥Young Gold+1e\ge \text{Young Gold}+1. Therefore Middle-aged Gold ≤0\le 0.
8
A count cannot be negative, so Middle-aged Gold =0=0.
Correct Answer: 0

Q28. Math Puzzles

MCQ · +3 / −1
Which of the following statements MUST be FALSE?
  1. A) The numbers of Gold and Platinum tickets bought by Young visitors were equal
  2. B) The numbers of Old and Middle-aged visitors buying Platinum tickets were equal
  3. C) The numbers of Old and Middle-aged visitors buying Economy tickets were equal
  4. D) The numbers of Middle-aged and Young visitors buying Gold tickets were equal
Correct Answer
C) The numbers of Old and Middle-aged visitors buying Economy tickets were equal
Step-by-step solution
📌 Core Concept
Express every group's counts through ee (Old Economy) and PP (total Platinum). A statement that MUST be false is one that leads to an impossible, non-integer value.
🔢 Step-by-Step Solution
1
Old =20=20, Middle-aged =40=40, Young =80=80. Old and Middle-aged together bought 55−38=1755-38=17 Economy tickets.
2
Let Old Economy =e=e. Then Old Gold =e=e, Old Platinum =20−2e=20-2e and Middle-aged Economy =17−e=17-e.
3
Option C says Old and Middle-aged Economy are equal: e=17−ee=17-e, so e=8.5e=8.5. This is not a whole number, so C is impossible.
Why Not the Others
A: Young Platinum =P/2=P/2 and Young Gold =42−P/2=42-P/2 are equal at P=42P=42. Then ee can be any value from 0 to 10 with all counts non-negative, so A is possible.
B: It is achieved in question 25, with 8 Old and 8 Middle-aged Platinum and P=32P=32, so B is possible.
D: Middle-aged Gold =43−e−P/2=43-e-P/2 and Young Gold =42−P/2=42-P/2 are equal when e=1e=1. For example P=40P=40 works, so D is possible.
Correct Answer: C
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CAT 2018 Slot 2 DILR at a Glance

Sets in this section8
Questions in this section32
Section time60 minutes
Convenor IIMIIM Calcutta

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