A person paid income tax at the rate of R% for the first Rs 2 lakhs, and at the rate of (R + 10)% for income beyond Rs 2lakhs. If the total tax paid is (R + 5)% of the annual income, then what is the annual income?

Correct option is C

Given:

• Tax rate on first Rs 2 lakhs = R%
• Tax rate beyond Rs 2 lakhs = (R + 10)%
• Total tax paid = (R + 5)% of total annual income
• We are to find the annual income

Concept Used:
We calculate the tax paid in two parts:

1. Fixed part on Rs 2 lakhs
2. Variable part on income above Rs 2 lakhs
   Then equate the total tax to (R + 5)% of the total income.

Formula Used:
Total tax = (R% of 2,00,000) + ((R + 10)% of amount exceeding 2,00,000)
Also,
Total tax = (R + 5)% of total income

Solution:

Let the total income be X
Then income above 2,00,000 = X – 2,00,000

Total tax = $$\frac{R}{100} \times 2,00,000 + \frac{R + 10}{100} \times (X - 2,00,000)$$​​
Also,
Total tax = $$\frac{R + 5}{100} \times X$$​​

Equating both:

​$$\frac{R \times 2,00,000}{100} + \frac{(R + 10)(X - 2,00,000)}{100} = \frac{(R + 5) \times X}{100}$$​​

Multiply the whole equation by 100 to eliminate denominators:

R × 2,00,000 + (R + 10)(X – 2,00,000) = (R + 5) × X

Now expand and simplify:

R × 2,00,000 + (R + 10)X – (R + 10) × 2,00,000 = (R + 5) × X

Now compute the constants:

R × 2,00,000 – (R + 10) × 2,00,000 = –10 × 2,00,000 = –20,00,000

So:
(R + 10)X – (R + 5)X = 20,00,000
→ 5X = 20,00,000
→ X = 4,00,000

Correct Option: (C) Rs 4.0 lakhs