Akshay started a business, and he invested Rs. 60,000. After some months, Rajni came to join with him and invest Rs. 1,50,000. At the end of the year, the total profit was divided among them into the ratio 42 : 35. After how many months did Rajni join?

Correct option is D

Given:
Akshay's investment = Rs. 60,000
Rajni's investment = Rs. 1,50,000
Profit ratio at the end of the year = 42 : 35
Total time of business = 12 months
Formula Used:
Profit ∝ Investment × Time
Profit Ratio = (Investment_A × Time_A) : (Investment_R × Time_R)
Solution:
Let Rajni joined after x months.
Akshay's time = 12 months
Rajni's time = (12 − x) months
Using the profit ratio:
​$$\frac{60,000 \times 12}{1,50,000 \times (12 - x)} = \frac{42}{35} \\\Rightarrow \frac{720,000}{1,50,000 \times (12 - x)} = \frac{6}{5} \\\Rightarrow 6 \times 1,50,000 \times (12 - x) = 5 \times 720,000 \\\Rightarrow 9,00,000 \times (12 - x) = 36,00,000 \\\Rightarrow 12 - x = \frac{36,00,000}{9,00,000} = 4 \\\Rightarrow x = 12 - 4 = 8$$​​
Rajni joined the business after 8 months.