A began a business with ₹ 85,000. He was joined afterward by B with ₹ 42,500. For how much period does B join, if the profits at the end of the year are divided in the ratio 3 : 1?

Correct option is C

Given:
- A's initial investment = ₹ 85,000
- B's initial investment = ₹ 42,500
- The profit ratio at the end of the year = 3:1 (A:B)

Formula Used:
The profit share is proportional to the investment and the time for which it is invested:
Profit Share = Investment × Time

Solution:
Let the time period for which B invested money be t months.

- A's investment = ₹ 85,000 for 12 months.
- B's investment = ₹ 42,500 for t months.

The ratio of the profits is:
​$$(\frac{A's \ Share }{B's \ Share}) = \frac{3}{1}$$​​

The share of A and B will be proportional to the product of their investments and time. So:
​$$\frac{(85,000 × 12) }{ (42,500 × t) }= \frac{3 }{ 1}$$​​

​$$\frac{1,020,000 }{ (42,500 × t)} = 3$$​​

1,020,000 = 3 × 42,500 × t

1,020,000 = 127,500 × t

t = $$\frac{1,020,000 }{127,500}$$​ = 8 months
B joined the business for  8 months .