Correct option is B

Given:
In the year 2021, the profit percentage of Company A is 60% and for Company B is 45%.
It is also given that the expenditure of Company A = expenditure of Company B.
Formula Used:
​$$\text{Profit (\%)} = \left( \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \right) \times 100$$​​

Solution:
Let the common expenditure for both companies be E.
For Company A:
​$$60 = \left( \frac{\text{Income}_A - E}{E} \right) \times 100 \\\Rightarrow \frac{\text{Income}_A - E}{E} = 0.6 \\\Rightarrow \text{Income}_A = 1.6E$$​​
For Company B:
​$$45= \left( \frac{\text{Income}_B - E}{E} \right) \times 100 \\\Rightarrow \frac{\text{Income}_B - E}{E} = 0.45 \\\Rightarrow \text{Income}_B = 1.45E$$​​
​$$\text{Income}_A : \text{Income}_B = 1.6E : 1.45E = 1.6 : 1.45 = 32 : 29$$​​

Correct option is C

Given:
In the year 2022:
Profit % of Company A = 45%
Profit % of Company B = 30%
Income of Company B = ₹52 Crore
Expenditures of both Companies A and B are equal
Formula Used:
​$$\text{Profit (\%)} = \left( \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \right) \times 100$$​​
Solution:
Expenditure using Company B's data:
​$$30 = \left( \frac{52 - E}{E} \right) \times 100 \\\Rightarrow \frac{52 - E}{E} = 0.3 \\\Rightarrow 52 - E = 0.3E \\\Rightarrow 52 = 1.3E \\\Rightarrow E = \frac{52}{1.3} = ₹40 \text{ Crore}$$​​
Income of Company A using its profit percentage:
​$$45 = \left( \frac{\text{Income}_A - 40}{40} \right) \times 100 \\\Rightarrow \frac{\text{Income}_A - 40}{40} = 0.45 \\\Rightarrow \text{Income}_A - 40 = 18 \\\Rightarrow \text{Income}_A = ₹58 \text{ Crore}$$​​
Average Income of A and B = $$\frac{58 + 52}{2}$$​ = ₹55 Crore

Correct option is D

Given:
In the year 2023:
Profit % of Company B = 40%
Income of Company B = ₹70 Crores
Formula Used:
​$$\text{Profit (\%)} = \left( \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \right) \times 100$$​
Solution:
​$$40 = \left( \frac{70 - E}{E} \right) \times 100 \\\Rightarrow \frac{70 - E}{E} = 0.4 \\\Rightarrow 70 - E = 0.4E \\\Rightarrow 70 = 1.4E \\\Rightarrow E = \frac{70}{1.4} = ₹50 \text{ Crores}$$​​

Correct option is D

​

Given:
Profit % of Company A in 2019 = 25%
Profit % of Company A in 2023 = 50%
Ratio of expenditures of Company A in 2019 and 2023 = 4 : 3
Formula Used:
​$$\text{Profit (\%)} = \left( \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \right) \times 100$$​​
Solution:
Let expenditure of Company A in 2019 = 4x
Then expenditure in 2023 = 3x
Income in 2019:
​$$25 = \left( \frac{\text{Income}_{2019} - 4x}{4x} \right) \times 100 \\\Rightarrow \frac{\text{Income}_{2019} - 4x}{4x} = 0.25 \\\Rightarrow \text{Income}_{2019} = 1.25 \times 4x = 5x$$​​
Income in 2023:
​$$50 = \left( \frac{\text{Income}_{2023} - 3x}{3x} \right) \times 100 \\\Rightarrow \frac{\text{Income}_{2023} - 3x}{3x} = 0.5 \\\Rightarrow \text{Income}_{2023} = 1.5 \times 3x = 4.5x \\\\$$​​
​$$\text{Required \%} = \left( \frac{5x - 4.5x}{4.5x} \right) \times 100 \\= \left( \frac{0.5x}{4.5x} \right) \times 100 = \left( \frac{1}{9} \right) \times 100 = \frac{100}{9} \%$$

Correct option is A

Given:
Profit % of Company B in 2018 = 25%
Profit % of Company B in 2020 = 50%
Income of Company B in 2018 = Expenditure of Company B in 2020
Formula Used:
​$$\text{Profit (\%)} = \left( \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \right) \times 100$$​​
Solution:
Income in 2018 = I₁
Income in 2020 = I₂
Expenditure in 2020 = E₂
Given: I₁ = E₂
Profit % in 2020:

$$50 = \left( \frac{I_2 - E_2}{E_2} \right) \times 100 \Rightarrow \frac{I_2 - E_2}{E_2} = 0.5 \Rightarrow I_2 = 1.5 \times E_2\\$$​​
Since I₁ = E₂, 
=> I₂ = 1.5 × I₁
Required Ratio = I₁ : I₂ = I₁ : 1.5I₁ = 1 : 1.5 = 2 : 3