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CAT 2025 Lesson : Numerical Reasoning - Cases 1 & 2
1. Introduction
This topic has been very popular in CAT, and there have been one or more cases requiring Numerical Reasoning in CAT every year, for the last few years. The examiners are trying to move away from calculation-intensive questions and towards reasoning-based questions. Therefore, practising such questions is really important.In such questions, there can be a bit of arithmetic involved, but we will need to use logic to solve them. For instance, we may be told of a sports league, with scores for each match (win, draw, lose). We could have the final scores as well as some additional information, and we will need to find the outcome of each match from these. We can have questions with different currencies and exchange rates, variables of companies (sales, costs, factory production) or countries (literacy, import-export), etc. We can be given data in tables as well. However, the use if logic or clues given will be paramount to solve such cases.
Let us solve a few questions to get a clear picture.
2. Solved Examples
Example Case 1
Read the following and answer the questions that follow
Some children were taking free throws at the basketball court in school during lunch break. Below are some facts about how many baskets these children shot.
i. Ganesh shot 8 baskets less than Ashish.
ii. Dhanraj and Ramesh together shot 37 baskets.
iii. Jugraj shot 8 baskets more than Dhanraj.
iv. Ashish shot 5 baskets more than Dhanraj.
v. Ashish and Ganesh together shot 40 baskets.
[CAT 2003 First Paper]
1) Which of the following statements is true?
(1) Ramesh shot 18 baskets and Dhanraj shot 19 baskets.
(2) Ganesh shot 24 baskets and Ashish shot 16 baskets.
(3) Jugraj shot 19 baskets and Dhanraj shot 27 baskets.
(4) Dhanraj shot 11 baskets and Ashish shot 16 baskets.
2) Which of the following statements is true?
(1) Dhanraj and Jugraj together shot 46 baskets.
(2) Ganesh shot 18 baskets and Ramesh shot 21 baskets.
(3) Dhanraj shot 3 more baskets than Ramesh.
(4) Ramesh and Jugraj together shot 29 baskets.
Let the children Ashish, Dhanraj, Ganesh, Jugraj & Ramesh be A, D, G, J & R respectively. Conditions given:
1. A = G + 8
2. R = 37 - D
3. J = D + 8
4. A = D + 5
5. A + G = 40
From 1 & 4,
G + 8 = D + 5
G = D - 3 -----(6)
From 5, substituting 4 & 6
A + G = 40
D + 5 + D - 3 = 40
2D = 38
D = 19 -----(7)
Substituting 7 in the conditions given to find the number of baskets the children shot
A = 24
G = 16
J = 27
R = 18
Option (4) satisfies the given conditions. Hence, option (1) is the answer.
2)
Let the children Ashish, Dhanraj, Ganesh, Jugraj & Ramesh be A, D, G, J & R respectively.
Conditions given:
1, A = G + 8
2, R = 37 – D
3, J = D + 8
4, A = D + 5
5, A + G = 40
From 1 & 4,
G + 8 = D + 5
G = D – 3 -----(6)
From 5, substituting 4 & 6
A + G = 40
D + 5 + D - 3 = 40
2D = 38
D = 19 -----(7)
Substituting 7 in the conditions given to find the number of baskets the children shot.
A = 24
G = 16
J = 27
R = 18
Option (1) satisfies the given condition. Hence, option 1 is the answer.
(1) Dhanraj and Jugraj together shot 46 baskets.
Answer:
1) (1) Ramesh shot 18 baskets and Dhanraj shot 19 baskets.
2) (1) Dhanraj and Jugraj together shot 46 baskets
Some children were taking free throws at the basketball court in school during lunch break. Below are some facts about how many baskets these children shot.
i. Ganesh shot 8 baskets less than Ashish.
ii. Dhanraj and Ramesh together shot 37 baskets.
iii. Jugraj shot 8 baskets more than Dhanraj.
iv. Ashish shot 5 baskets more than Dhanraj.
v. Ashish and Ganesh together shot 40 baskets.
[CAT 2003 First Paper]
1) Which of the following statements is true?
(1) Ramesh shot 18 baskets and Dhanraj shot 19 baskets.
(2) Ganesh shot 24 baskets and Ashish shot 16 baskets.
(3) Jugraj shot 19 baskets and Dhanraj shot 27 baskets.
(4) Dhanraj shot 11 baskets and Ashish shot 16 baskets.
2) Which of the following statements is true?
(1) Dhanraj and Jugraj together shot 46 baskets.
(2) Ganesh shot 18 baskets and Ramesh shot 21 baskets.
(3) Dhanraj shot 3 more baskets than Ramesh.
(4) Ramesh and Jugraj together shot 29 baskets.
Solution
1)Let the children Ashish, Dhanraj, Ganesh, Jugraj & Ramesh be A, D, G, J & R respectively. Conditions given:
1. A = G + 8
2. R = 37 - D
3. J = D + 8
4. A = D + 5
5. A + G = 40
From 1 & 4,
G + 8 = D + 5
G = D - 3 -----(6)
From 5, substituting 4 & 6
A + G = 40
D + 5 + D - 3 = 40
2D = 38
D = 19 -----(7)
Substituting 7 in the conditions given to find the number of baskets the children shot
A = 24
G = 16
J = 27
R = 18
Option (4) satisfies the given conditions. Hence, option (1) is the answer.
2)
Let the children Ashish, Dhanraj, Ganesh, Jugraj & Ramesh be A, D, G, J & R respectively.
Conditions given:
1, A = G + 8
2, R = 37 – D
3, J = D + 8
4, A = D + 5
5, A + G = 40
From 1 & 4,
G + 8 = D + 5
G = D – 3 -----(6)
From 5, substituting 4 & 6
A + G = 40
D + 5 + D - 3 = 40
2D = 38
D = 19 -----(7)
Substituting 7 in the conditions given to find the number of baskets the children shot.
A = 24
G = 16
J = 27
R = 18
Option (1) satisfies the given condition. Hence, option 1 is the answer.
(1) Dhanraj and Jugraj together shot 46 baskets.
Answer:
1) (1) Ramesh shot 18 baskets and Dhanraj shot 19 baskets.
2) (1) Dhanraj and Jugraj together shot 46 baskets
Example Case 2
Read the following and answer the questions that follow
Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 and Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss.
[CAT 2008]
3) Which of the following cannot be true?
(1) At least two horses finished before Spotted
(2) Red finished last
(3) There were three horses between Black and Spotted
(4) There were three horses between White and Red
(5) Grey came in second
4) Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?
(1) Spotted came in first
(2) Red finished last
(3) White came in second
(4) Black came in second
(5) There was one horse between Black and White
Amount invested by Raju = 3000 + 2000 + 1000 = 6000
Let us form a table for the position of horses where Raju will end up with no profit and no loss.
Raju bets Rs. 3000, Rs. 2000, Rs. 1000 on Red, White and Black.
3 scenarios are possible, where Raju will end up with no profit and no loss.
In Scenario 1, Red/Black can only be in positions 4 and 5.
In Scenario 2, White can only be in either position 4 or 5.
In Scenario 3, Red can only be in either position 4 or 5.
3)
Let us consider the options.
Option (1) – possible in all the scenarios.
Option (2) – possible in scenario 1.
Option (3) – possible in scenario 1, if Black finishes 3rd and Spotted finishes 1st.
Option (4) – not possible in any of the three scenarios. Therefore, this is the answer.
Option (5) – possible in scenario 3.
Answer: (4) There were three horses between White and Red
4)
If Grey came 4th, then scenario 1 is not possible and we can completely fill the remaining table for the other 2 scenarios.
From the above table, only option (3) is not possible, which is the correct choice.
Answer: (3) White came in second
Answer:
3) (4) There were three horses between White and Red
4) (3) White came in second
Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 and Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss.
[CAT 2008]
3) Which of the following cannot be true?
(1) At least two horses finished before Spotted
(2) Red finished last
(3) There were three horses between Black and Spotted
(4) There were three horses between White and Red
(5) Grey came in second
4) Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?
(1) Spotted came in first
(2) Red finished last
(3) White came in second
(4) Black came in second
(5) There was one horse between Black and White
Solution
Case SolutionAmount invested by Raju = 3000 + 2000 + 1000 = 6000
Let us form a table for the position of horses where Raju will end up with no profit and no loss.
Raju bets Rs. 3000, Rs. 2000, Rs. 1000 on Red, White and Black.
3 scenarios are possible, where Raju will end up with no profit and no loss.
In Scenario 1, Red/Black can only be in positions 4 and 5.
In Scenario 2, White can only be in either position 4 or 5.
In Scenario 3, Red can only be in either position 4 or 5.
3)
Let us consider the options.
Option (1) – possible in all the scenarios.
Option (2) – possible in scenario 1.
Option (3) – possible in scenario 1, if Black finishes 3rd and Spotted finishes 1st.
Option (4) – not possible in any of the three scenarios. Therefore, this is the answer.
Option (5) – possible in scenario 3.
Answer: (4) There were three horses between White and Red
4)
If Grey came 4th, then scenario 1 is not possible and we can completely fill the remaining table for the other 2 scenarios.
From the above table, only option (3) is not possible, which is the correct choice.
Answer: (3) White came in second
Answer:
3) (4) There were three horses between White and Red
4) (3) White came in second
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