CAT 2023 Slot 2 DILR Question & Solution
Data Set
Odsville has five firms - Alfloo, Bzygoo, Czechy, Drjbna and Elavalaki. Each of these firms was founded in some year and also closed down a few years later.
Each firm raised Rs. 1 crore in its first and last year of existence. The amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down. No firm raised the same amount of money in two consecutive years. Each annual increase and decrease was either by Rs. 1 crore or by Rs. 2 crores. The table below provides partial information about the five firms.
Question 1
For which firm(s) can the amounts raised by them be concluded with certainty in each year?
Solution:
In this set, we are told that the amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down and no firm raised the same amount of money in two consecutive years.
The increase or decrease can be $\pm\ 1$ or $\pm\ 2$. => (1)
We are also told that each firm raised Rs. 1 crore in its first and last year of existence
Consider A:
It raised money for 8 years
=> The raising pattern looks like follows:
1, a, b, c, d, e, f, 1 => where a, b, c,..,, f are the unknown amounts raised.
Also a + b + c + d + e + f = 21 - 2 = 19.
We can observe that 19/6 is slightly greater than 3 => The average amount raised should be around 3.
If a = 3 and f = 3 => b + c + d + e = 13 (not possible) as the minimum case would be (4, 5, 6, 4) => Not possible.
If a = 3 and f = 2 => b + c + d + e = 14 (not possible) as the minimum case would be (4, 5, 4, 3) => Not possible.
=> a = 2 and f = 2 => b + c + d + e = 15 the minimum case is (3, 4, 5, 3) or (3, 5, 4, 3) which gives a sum of 15.
So, the possible cases for A are:
Consider B:
The patterns looks as follows:
1, a, b, 1
If a = 2, b has to be equal to 3 to satisfy (1)
if a = 3, b has to be equal to 2 to satisfy (1)
=> The possible cases for B are:
Consider C:
The pattern looks as follows:
1, ..., 1
Let us assume there are 2 gaps between => a + b = 7 (Not possible) as maximum case would be 1, 3, 2, 1
Let us assume there are 3 gaps between => a + b + c = 7, the minimum case possible is 1, 2, 3, 2, 1 => Satisfies.
Now, if there are 4 gaps => a + b + c + d = 7 => The average value is 7/4 which is less than 2 => Not possible.
=> The possible cases for C are:
Consider D:
The pattern looks as follows:
1, a, b, c, 1
=> a + b + c = 8
When a = 2 and c = 2 => b = 4 => 2, 4, 2 => Satisfies.
When a = 2 and c = 3, b should be 3 (Not satisfying (1))
When a = 3 and c = 3, b should be 2 (Not satisfying (1))
=> The possible cases for D are:
Consider E:
The pattern looks as follows:
1,.....,1
For 1 or 2 gaps, we can't get a sum of 11.
Assume 3 gaps => a + b + c = 11, the maximum case is 3, 5, 3 => Satisfies.
Now, assume 4 gaps
=> a + b + c + d = 11, the minimum case is 2, 3, 4, 2 or 2, 4, 3, 2 which satisfies (1) and 2 + 3 + 4 + 2 = 11.
=> The possible cases for E are:
In summary, the possible cases for all 5 companies is:
We see that only for C and D, we can conclude the amounts raised with certainty.
Question 2
What best can be concluded about the total amount of money raised in 2015?
Solution:
In this set, we are told that the amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down and no firm raised the same amount of money in two consecutive years.
The increase or decrease can be $\pm\ 1$ or $\pm\ 2$. => (1)
We are also told that each firm raised Rs. 1 crore in its first and last year of existence
Consider A:
It raised money for 8 years
=> The raising pattern looks like follows:
1, a, b, c, d, e, f, 1 => where a, b, c,..,, f are the unknown amounts raised.
Also a + b + c + d + e + f = 21 - 2 = 19.
We can observe that 19/6 is slightly greater than 3 => The average amount raised should be around 3.
If a = 3 and f = 3 => b + c + d + e = 13 (not possible) as the minimum case would be (4, 5, 6, 4) => Not possible.
If a = 3 and f = 2 => b + c + d + e = 14 (not possible) as the minimum case would be (4, 5, 4, 3) => Not possible.
=> a = 2 and f = 2 => b + c + d + e = 15 the minimum case is (3, 4, 5, 3) or (3, 5, 4, 3) which gives a sum of 15.
So, the possible cases for A are:
Consider B:
The patterns looks as follows:
1, a, b, 1
If a = 2, b has to be equal to 3 to satisfy (1)
if a = 3, b has to be equal to 2 to satisfy (1)
=> The possible cases for B are:
Consider C:
The pattern looks as follows:
1, ..., 1
Let us assume there are 2 gaps between => a + b = 7 (Not possible) as maximum case would be 1, 3, 2, 1
Let us assume there are 3 gaps between => a + b + c = 7, the minimum case possible is 1, 2, 3, 2, 1 => Satisfies.
Now, if there are 4 gaps => a + b + c + d = 7 => The average value is 7/4 which is less than 2 => Not possible.
=> The possible cases for C are:
Consider D:
The pattern looks as follows:
1, a, b, c, 1
=> a + b + c = 8
When a = 2 and c = 2 => b = 4 => 2, 4, 2 => Satisfies.
When a = 2 and c = 3, b should be 3 (Not satisfying (1))
When a = 3 and c = 3, b should be 2 (Not satisfying (1))
=> The possible cases for D are:
Consider E:
The pattern looks as follows:
1,.....,1
For 1 or 2 gaps, we can't get a sum of 11.
Assume 3 gaps => a + b + c = 11, the maximum case is 3, 5, 3 => Satisfies.
Now, assume 4 gaps
=> a + b + c + d = 11, the minimum case is 2, 3, 4, 2 or 2, 4, 3, 2 which satisfies (1) and 2 + 3 + 4 + 2 = 11.
=> The possible cases for E are:
In summary, the possible cases for all 5 companies is:
Money raised in 2015 is 2 + 1 + 3 + 1 + 0/1 = 7 or 8.
Question 3
What is the largest possible total amount of money (in Rs. crores) that could have been raised in 2013?
Solution:
In this set, we are told that the amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down and no firm raised the same amount of money in two consecutive years.
The increase or decrease can be $\pm\ 1$ or $\pm\ 2$. => (1)
We are also told that each firm raised Rs. 1 crore in its first and last year of existence
Consider A:
It raised money for 8 years
=> The raising pattern looks like follows:
1, a, b, c, d, e, f, 1 => where a, b, c,..,, f are the unknown amounts raised.
Also a + b + c + d + e + f = 21 - 2 = 19.
We can observe that 19/6 is slightly greater than 3 => The average amount raised should be around 3.
If a = 3 and f = 3 => b + c + d + e = 13 (not possible) as the minimum case would be (4, 5, 6, 4) => Not possible.
If a = 3 and f = 2 => b + c + d + e = 14 (not possible) as the minimum case would be (4, 5, 4, 3) => Not possible.
=> a = 2 and f = 2 => b + c + d + e = 15 the minimum case is (3, 4, 5, 3) or (3, 5, 4, 3) which gives a sum of 15.
So, the possible cases for A are:
Consider B:
The patterns looks as follows:
1, a, b, 1
If a = 2, b has to be equal to 3 to satisfy (1)
if a = 3, b has to be equal to 2 to satisfy (1)
=> The possible cases for B are:
Consider C:
The pattern looks as follows:
1, ..., 1
Let us assume there are 2 gaps between => a + b = 7 (Not possible) as maximum case would be 1, 3, 2, 1
Let us assume there are 3 gaps between => a + b + c = 7, the minimum case possible is 1, 2, 3, 2, 1 => Satisfies.
Now, if there are 4 gaps => a + b + c + d = 7 => The average value is 7/4 which is less than 2 => Not possible.
=> The possible cases for C are:
Consider D:
The pattern looks as follows:
1, a, b, c, 1
=> a + b + c = 8
When a = 2 and c = 2 => b = 4 => 2, 4, 2 => Satisfies.
When a = 2 and c = 3, b should be 3 (Not satisfying (1))
When a = 3 and c = 3, b should be 2 (Not satisfying (1))
=> The possible cases for D are:
Consider E:
The pattern looks as follows:
1,.....,1
For 1 or 2 gaps, we can't get a sum of 11.
Assume 3 gaps => a + b + c = 11, the maximum case is 3, 5, 3 => Satisfies.
Now, assume 4 gaps
=> a + b + c + d = 11, the minimum case is 2, 3, 4, 2 or 2, 4, 3, 2 which satisfies (1) and 2 + 3 + 4 + 2 = 11.
=> The possible cases for E are:
In summary, the possible cases for all 5 companies is:
Maximum money raised in 2013 is 5 + 3 + 1 + 4 + 4 = 17.
Question 4
If Elavalaki raised Rs. 3 crores in 2013, then what is the smallest possible total amount of money (in Rs. crores) that could have been raised by all the companies in 2012?
Solution:
In this set, we are told that the amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down and no firm raised the same amount of money in two consecutive years.
The increase or decrease can be $\pm\ 1$ or $\pm\ 2$. => (1)
We are also told that each firm raised Rs. 1 crore in its first and last year of existence
Consider A:
It raised money for 8 years
=> The raising pattern looks like follows:
1, a, b, c, d, e, f, 1 => where a, b, c,..,, f are the unknown amounts raised.
Also a + b + c + d + e + f = 21 - 2 = 19.
We can observe that 19/6 is slightly greater than 3 => The average amount raised should be around 3.
If a = 3 and f = 3 => b + c + d + e = 13 (not possible) as the minimum case would be (4, 5, 6, 4) => Not possible.
If a = 3 and f = 2 => b + c + d + e = 14 (not possible) as the minimum case would be (4, 5, 4, 3) => Not possible.
=> a = 2 and f = 2 => b + c + d + e = 15 the minimum case is (3, 4, 5, 3) or (3, 5, 4, 3) which gives a sum of 15.
So, the possible cases for A are:
Consider B:
The patterns looks as follows:
1, a, b, 1
If a = 2, b has to be equal to 3 to satisfy (1)
if a = 3, b has to be equal to 2 to satisfy (1)
=> The possible cases for B are:
Consider C:
The pattern looks as follows:
1, ..., 1
Let us assume there are 2 gaps between => a + b = 7 (Not possible) as maximum case would be 1, 3, 2, 1
Let us assume there are 3 gaps between => a + b + c = 7, the minimum case possible is 1, 2, 3, 2, 1 => Satisfies.
Now, if there are 4 gaps => a + b + c + d = 7 => The average value is 7/4 which is less than 2 => Not possible.
=> The possible cases for C are:
Consider D:
The pattern looks as follows:
1, a, b, c, 1
=> a + b + c = 8
When a = 2 and c = 2 => b = 4 => 2, 4, 2 => Satisfies.
When a = 2 and c = 3, b should be 3 (Not satisfying (1))
When a = 3 and c = 3, b should be 2 (Not satisfying (1))
=> The possible cases for D are:
Consider E:
The pattern looks as follows:
1,.....,1
For 1 or 2 gaps, we can't get a sum of 11.
Assume 3 gaps => a + b + c = 11, the maximum case is 3, 5, 3 => Satisfies.
Now, assume 4 gaps
=> a + b + c + d = 11, the minimum case is 2, 3, 4, 2 or 2, 4, 3, 2 which satisfies (1) and 2 + 3 + 4 + 2 = 11.
=> The possible cases for E are:
In summary, the possible cases for all 5 companies is:
Given that E raised 3 in 2013 => in 2012 he could have raised a minimum of 4 crores.
=> Minimum amount is 4 + 1 + 0 + 2 + 4 = 11.
Question 5
If the total amount of money raised in 2014 is Rs. 12 crores, then which of the following is not possible?
Solution:
In this set, we are told that the amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down and no firm raised the same amount of money in two consecutive years.
The increase or decrease can be $\pm\ 1$ or $\pm\ 2$. => (1)
We are also told that each firm raised Rs. 1 crore in its first and last year of existence
Consider A:
It raised money for 8 years
=> The raising pattern looks like follows:
1, a, b, c, d, e, f, 1 => where a, b, c,..,, f are the unknown amounts raised.
Also a + b + c + d + e + f = 21 - 2 = 19.
We can observe that 19/6 is slightly greater than 3 => The average amount raised should be around 3.
If a = 3 and f = 3 => b + c + d + e = 13 (not possible) as the minimum case would be (4, 5, 6, 4) => Not possible.
If a = 3 and f = 2 => b + c + d + e = 14 (not possible) as the minimum case would be (4, 5, 4, 3) => Not possible.
=> a = 2 and f = 2 => b + c + d + e = 15 the minimum case is (3, 4, 5, 3) or (3, 5, 4, 3) which gives a sum of 15.
So, the possible cases for A are:
Consider B:
The patterns looks as follows:
1, a, b, 1
If a = 2, b has to be equal to 3 to satisfy (1)
if a = 3, b has to be equal to 2 to satisfy (1)
=> The possible cases for B are:
Consider C:
The pattern looks as follows:
1, ..., 1
Let us assume there are 2 gaps between => a + b = 7 (Not possible) as maximum case would be 1, 3, 2, 1
Let us assume there are 3 gaps between => a + b + c = 7, the minimum case possible is 1, 2, 3, 2, 1 => Satisfies.
Now, if there are 4 gaps => a + b + c + d = 7 => The average value is 7/4 which is less than 2 => Not possible.
=> The possible cases for C are:
Consider D:
The pattern looks as follows:
1, a, b, c, 1
=> a + b + c = 8
When a = 2 and c = 2 => b = 4 => 2, 4, 2 => Satisfies.
When a = 2 and c = 3, b should be 3 (Not satisfying (1))
When a = 3 and c = 3, b should be 2 (Not satisfying (1))
=> The possible cases for D are:
Consider E:
The pattern looks as follows:
1,.....,1
For 1 or 2 gaps, we can't get a sum of 11.
Assume 3 gaps => a + b + c = 11, the maximum case is 3, 5, 3 => Satisfies.
Now, assume 4 gaps
=> a + b + c + d = 11, the minimum case is 2, 3, 4, 2 or 2, 4, 3, 2 which satisfies (1) and 2 + 3 + 4 + 2 = 11.
=> The possible cases for E are:
In summary, the possible cases for all 5 companies is:
Given that total amount raised in 2014 is 12
=> 3 + 3/2 + 2 + 2 + 1/2 = 12 =>
=> possible case is 3 + 3 + 2 + 2 + 2 = 12.
A) In 2013, B raised 2 crores and E also raised 3/4 crores => Not Possible.
B) In 2013, A could have raised 5/4 and D raised 4 => Possible.
C) In 2014, A raised 3 and B raised 3 => Possible.
D) In 2014, B raised 3 where as E raised 2 => 3 > 2 => Possible.