CAT 2025 Slot 2 DILR Question & Solution
Data Set
The Sustainability Index (SI) of a country at a point in time is an integer between 1 and 100. This question is related to SI of six countries - A, B, C, D, E, and F - at three different points in time - 2016, 2020, and 2024. The plot represents the exact changes in their SI, with X-coordinate representing % increase in 2020 from 2016, i.e., (SI in 2020 minus SI in 2016) / (SI in 2016), and Y-coordinate representing % increase in 2024 from 2020. At any point in time, the country with highest SI is ranked 1, while the country with the lowest SI is ranked 6. The following additional facts are known.
1. In 2016, B, C, E, and A had ranks 1, 2, 3, and 4 respectively.
2. F had lower SI than any other country in 2016, 2020, and 2024.
3. In 2024, E was the only country with SI of 90.
4. The range of SI of the six countries was 60 in 2016 as well as in 2024.
Question 1
What was the SI of E in 2016?
Solution:
Let us assume the values of A, B, C, D, E and F to be a, b, c, d, e and f in 2016.
For A:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)a$$ = $$\dfrac{5}{4}a$$
Value in 2024 = increased by 50% from 2020 = $$\left(1\ +\ \dfrac{50}{100}\right)\times\ \dfrac{5a}{4}$$ = $$\dfrac{15}{8}a$$
For B:
Value in 2020 = decreased by 25% from 2016 = $$\left(1\ -\ \dfrac{25}{100}\right)b$$ = $$\dfrac{3}{4}b$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ \dfrac{3b}{4}$$ = $$\dfrac{9}{16}b$$
For C:
Value in 2020 = decreased by 20% from 2016 = $$\left(1\ -\ \dfrac{20}{100}\right)c$$ = $$\dfrac{4}{5}c$$
Value in 2024 = increased by 40% from 2020 = $$\left(1\ +\ \dfrac{40}{100}\right)\times\ \dfrac{4c}{5}$$ = $$\dfrac{28}{25}c$$
For D:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)d$$ = $$2d$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ 2d$$ = $$\dfrac{12}{5}d$$
For E:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)e$$ = $$\dfrac{5}{4}e$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ \dfrac{5e}{4}$$ = $$\dfrac{3}{2}e$$
For F:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)f$$ = $$2f$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ 2f$$ = $$\dfrac{3}{2}f$$
We are given that the value of E is 90 in 2024. We can calculate the value of e as,
$$\dfrac{3e}{2}\ =\ 90$$
$$e\ =\ 60$$
The value of E in 2016 is 60, and its value in 2020 can be calculated as,
$$\dfrac{5e}{4}\ =\ \dfrac{5}{4}\times\ 60\ =\ 75$$
We are given that B, C, E, and A had ranks 1, 2, 3, and 4, respectively, in 2016, and F has the lowest rank(6) in all the years.
Putting the values in the table, we get,
SI of E in 2016 is 60.
Hence, the correct answer is 60.
Question 2
What was the SI of F in 2020?
Solution:
Let us assume the values of A, B, C, D, E and F to be a, b, c, d, e and f in 2016.
For A:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)a$$ = $$\dfrac{5}{4}a$$
Value in 2024 = increased by 50% from 2020 = $$\left(1\ +\ \dfrac{50}{100}\right)\times\ \dfrac{5a}{4}$$ = $$\dfrac{15}{8}a$$
For B:
Value in 2020 = decreased by 25% from 2016 = $$\left(1\ -\ \dfrac{25}{100}\right)b$$ = $$\dfrac{3}{4}b$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ \dfrac{3b}{4}$$ = $$\dfrac{9}{16}b$$
For C:
Value in 2020 = decreased by 20% from 2016 = $$\left(1\ -\ \dfrac{20}{100}\right)c$$ = $$\dfrac{4}{5}c$$
Value in 2024 = increased by 40% from 2020 = $$\left(1\ +\ \dfrac{40}{100}\right)\times\ \dfrac{4c}{5}$$ = $$\dfrac{28}{25}c$$
For D:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)d$$ = $$2d$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ 2d$$ = $$\dfrac{12}{5}d$$
For E:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)e$$ = $$\dfrac{5}{4}e$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ \dfrac{5e}{4}$$ = $$\dfrac{3}{2}e$$
For F:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)f$$ = $$2f$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ 2f$$ = $$\dfrac{3}{2}f$$
We are given that the value of E is 90 in 2024. We can calculate the value of e as,
$$\dfrac{3e}{2}\ =\ 90$$
$$e\ =\ 60$$
The value of E in 2016 is 60, and its value in 2020 can be calculated as,
$$\dfrac{5e}{4}\ =\ \dfrac{5}{4}\times\ 60\ =\ 75$$
We are given that B, C, E, and A had ranks 1, 2, 3, and 4, respectively, in 2016, and F has the lowest rank(6) in all the years.
Putting the values in the table, we get,
We know that F is ranked 6 and B is ranked 1 in 2016, and we are also given that the range of SI in 2016 and 2024 is 60. This means that the value of b-f = 60, and we can conclude that b is definitely greater than 60, as we know that f is an integer greater than 0.
We know that all the SI values in all the years are integers, and for SI of B to be an integer in 2024, the value of b must be a multiple of 16. The multiples of 16 that are greater than 60 and less than 100 are 64, 80 and 96.
CASE 1: b = 64
If b = 64, then the value of f becomes 4, as we know b - f = 60.
If f = 4, then the value of F in 2024 is 3f/2 = 12/2 = 6.
We already know the value of E in 2024 is 90.
In that case, the range of SI in 2024 is at least 90 - 6 = 84, but we are given that the range is 60 in 2024, which is not possible in this case.
So, we can eliminate this case.
CASE 2: b = 96
If b = 96, then the value of f becomes 36, as we know b - f = 60.
If f = 36, then the value of F in 2024 is 3f/2 = 108/2 = 54.
The value of B in 2024 = 9b/16 = (9 * 96)/16 = 54
We calculated the value of B and F to be 54 in 2024, which is not possible, as we are given that F had a lower SI than any other country in 2024, and we obtained its value to be the same as B.
So, we can eliminate this case.
CASE 3: b = 80
If b = 80, then the value of f becomes 20, as we know b - f = 60.
If f = 20, then the value of F in 2024 is 3f/2 = 60/2 = 30.
The value of B in 2024 = 9b/16 = (9 * 80)/16 = 45
We can see that no condition is violated in this case, as F is less than B in 2024, and the range in 2024 from the obtained values as of now is 90 - 30 = 60.
So, we can conclude that the value of b = 80 and f = 20.
Substituting the values in the table, we get,
So, SI of F in 2020 is 40.
Hence, the correct answer is 40.
Question 3
What was the SI of C in 2024?
Solution:
Let us assume the values of A, B, C, D, E and F to be a, b, c, d, e and f in 2016.
For A:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)a$$ = $$\dfrac{5}{4}a$$
Value in 2024 = increased by 50% from 2020 = $$\left(1\ +\ \dfrac{50}{100}\right)\times\ \dfrac{5a}{4}$$ = $$\dfrac{15}{8}a$$
For B:
Value in 2020 = decreased by 25% from 2016 = $$\left(1\ -\ \dfrac{25}{100}\right)b$$ = $$\dfrac{3}{4}b$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ \dfrac{3b}{4}$$ = $$\dfrac{9}{16}b$$
For C:
Value in 2020 = decreased by 20% from 2016 = $$\left(1\ -\ \dfrac{20}{100}\right)c$$ = $$\dfrac{4}{5}c$$
Value in 2024 = increased by 40% from 2020 = $$\left(1\ +\ \dfrac{40}{100}\right)\times\ \dfrac{4c}{5}$$ = $$\dfrac{28}{25}c$$
For D:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)d$$ = $$2d$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ 2d$$ = $$\dfrac{12}{5}d$$
For E:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)e$$ = $$\dfrac{5}{4}e$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ \dfrac{5e}{4}$$ = $$\dfrac{3}{2}e$$
For F:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)f$$ = $$2f$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ 2f$$ = $$\dfrac{3}{2}f$$
We are given that the value of E is 90 in 2024. We can calculate the value of e as,
$$\dfrac{3e}{2}\ =\ 90$$
$$e\ =\ 60$$
The value of E in 2016 is 60, and its value in 2020 can be calculated as,
$$\dfrac{5e}{4}\ =\ \dfrac{5}{4}\times\ 60\ =\ 75$$
We are given that B, C, E, and A had ranks 1, 2, 3, and 4, respectively, in 2016, and F has the lowest rank(6) in all the years.
Putting the values in the table, we get,
We know that F is ranked 6 and B is ranked 1 in 2016, and we are also given that the range of SI in 2016 and 2024 is 60. This means that the value of b-f = 60, and we can conclude that b is definitely greater than 60, as we know that f is an integer greater than 0.
We know that all the SI values in all the years are integers, and for SI of B to be an integer in 2024, the value of b must be a multiple of 16. The multiples of 16 that are greater than 60 and less than 100 are 64, 80 and 96.
CASE 1: b = 64
If b = 64, then the value of f becomes 4, as we know b - f = 60.
If f = 4, then the value of F in 2024 is 3f/2 = 12/2 = 6.
We already know the value of E in 2024 is 90.
In that case, the range of SI in 2024 is at least 90 - 6 = 84, but we are given that the range is 60 in 2024, which is not possible in this case.
So, we can eliminate this case.
CASE 2: b = 96
If b = 96, then the value of f becomes 36, as we know b - f = 60.
If f = 36, then the value of F in 2024 is 3f/2 = 108/2 = 54.
The value of B in 2024 = 9b/16 = (9 * 96)/16 = 54
We calculated the value of B and F to be 54 in 2024, which is not possible, as we are given that F had a lower SI than any other country in 2024, and we obtained its value to be the same as B.
So, we can eliminate this case.
CASE 3: b = 80
If b = 80, then the value of f becomes 20, as we know b - f = 60.
If f = 20, then the value of F in 2024 is 3f/2 = 60/2 = 30.
The value of B in 2024 = 9b/16 = (9 * 80)/16 = 45
We can see that no condition is violated in this case, as F is less than B in 2024, and the range in 2024 from the obtained values as of now is 90 - 30 = 60.
So, we can conclude that the value of b = 80 and f = 20.
Substituting the values in the table, we get,
We know that C is ranked 2nd in 2016, and we know the values of 1st and 3rd to be 80 and 60. We also know that SI of C has to be integers in 2016, 2020 and 2024. So, for the value of SI of C to be an integer in 2024, the value of c has to be a multiple of 25.
So, we know that c lies between 80 and 60, and is also a multiple of 25. The only multiple of 25 that lies between 60 and 80 is 75. So, the value of c has to be 75, and the value of C in 2024 can be calculated as,
C in 2024 = 28c/25 = (28 * 75)/25 = 84.
Hence, the correct answer is 84.
Question 4
What was the SI of B in 2024?
Solution:
Let us assume the values of A, B, C, D, E and F to be a, b, c, d, e and f in 2016.
For A:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)a$$ = $$\dfrac{5}{4}a$$
Value in 2024 = increased by 50% from 2020 = $$\left(1\ +\ \dfrac{50}{100}\right)\times\ \dfrac{5a}{4}$$ = $$\dfrac{15}{8}a$$
For B:
Value in 2020 = decreased by 25% from 2016 = $$\left(1\ -\ \dfrac{25}{100}\right)b$$ = $$\dfrac{3}{4}b$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ \dfrac{3b}{4}$$ = $$\dfrac{9}{16}b$$
For C:
Value in 2020 = decreased by 20% from 2016 = $$\left(1\ -\ \dfrac{20}{100}\right)c$$ = $$\dfrac{4}{5}c$$
Value in 2024 = increased by 40% from 2020 = $$\left(1\ +\ \dfrac{40}{100}\right)\times\ \dfrac{4c}{5}$$ = $$\dfrac{28}{25}c$$
For D:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)d$$ = $$2d$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ 2d$$ = $$\dfrac{12}{5}d$$
For E:
Value in 2020 = increased by 25% from 2016 = $$\left(1\ +\ \dfrac{25}{100}\right)e$$ = $$\dfrac{5}{4}e$$
Value in 2024 = increased by 20% from 2020 = $$\left(1\ +\ \dfrac{20}{100}\right)\times\ \dfrac{5e}{4}$$ = $$\dfrac{3}{2}e$$
For F:
Value in 2020 = increased by 100% from 2016 = $$\left(1\ +\ \dfrac{100}{100}\right)f$$ = $$2f$$
Value in 2024 = decreased by 25% from 2020 = $$\left(1\ -\ \dfrac{25}{100}\right)\times\ 2f$$ = $$\dfrac{3}{2}f$$
We are given that the value of E is 90 in 2024. We can calculate the value of e as,
$$\dfrac{3e}{2}\ =\ 90$$
$$e\ =\ 60$$
The value of E in 2016 is 60, and its value in 2020 can be calculated as,
$$\dfrac{5e}{4}\ =\ \dfrac{5}{4}\times\ 60\ =\ 75$$
We are given that B, C, E, and A had ranks 1, 2, 3, and 4, respectively, in 2016, and F has the lowest rank(6) in all the years.
Putting the values in the table, we get,
We know that F is ranked 6 and B is ranked 1 in 2016, and we are also given that the range of SI in 2016 and 2024 is 60. This means that the value of b-f = 60, and we can conclude that b is definitely greater than 60, as we know that f is an integer greater than 0.
We know that all the SI values in all the years are integers, and for SI of B to be an integer in 2024, the value of b must be a multiple of 16. The multiples of 16 that are greater than 60 and less than 100 are 64, 80 and 96.
CASE 1: b = 64
If b = 64, then the value of f becomes 4, as we know b - f = 60.
If f = 4, then the value of F in 2024 is 3f/2 = 12/2 = 6.
We already know the value of E in 2024 is 90.
In that case, the range of SI in 2024 is at least 90 - 6 = 84, but we are given that the range is 60 in 2024, which is not possible in this case.
So, we can eliminate this case.
CASE 2: b = 96
If b = 96, then the value of f becomes 36, as we know b - f = 60.
If f = 36, then the value of F in 2024 is 3f/2 = 108/2 = 54.
The value of B in 2024 = 9b/16 = (9 * 96)/16 = 54
We calculated the value of B and F to be 54 in 2024, which is not possible, as we are given that F had a lower SI than any other country in 2024, and we obtained its value to be the same as B.
So, we can eliminate this case.
CASE 3: b = 80
If b = 80, then the value of f becomes 20, as we know b - f = 60.
If f = 20, then the value of F in 2024 is 3f/2 = 60/2 = 30.
The value of B in 2024 = 9b/16 = (9 * 80)/16 = 45
We can see that no condition is violated in this case, as F is less than B in 2024, and the range in 2024 from the obtained values as of now is 90 - 30 = 60.
So, we can conclude that the value of b = 80 and f = 20.
Substituting the values in the table, we get,
So, SI of B in 2024 is 45.
Hence, the correct answer is option D.