P, Q and R enter into a partnership by investing their capitals in the ratio of \(\rm \frac{2}{5}:\frac{3}{4}:\frac{5}{8}\). after 4 months, P increased his capital by 50%, but Q decreased his capital by 20%. What is the share of Q in the total profit of ₹28,210 at the end of a year?

Given:

Initial ratio of investments by P, Q, and R = 2/5 : 3/4 : 5/8

After 4 months, P increased his capital by 50% and Q decreased his capital by 20%

Total profit at the end of the year = ₹28,210

Formula used:

Share of profit = (Capital × Time) / Total Capital × Total Time

Calculation:

Let's convert the ratios to a common base for easier comparison:

P's initial investment = 2/5 = 16/40

Q's initial investment = 3/4 = 30/40

R's initial investment = 5/8 = 25/40

Total months = 12

For the first 4 months:

P's capital = 16 units, Q's capital = 30 units, R's capital = 25 units

For the next 8 months:

P increased his capital by 50% ⇒ New P's capital = 16 + 0.5 × 16 = 24 units

Q decreased his capital by 20% ⇒ New Q's capital = 30 - 0.2 × 30 = 24 units

R's capital remains the same = 25 units

Total investment in terms of units:

P's contribution = 16 × 4 + 24 × 8 = 256 units

Q's contribution = 30 × 4 + 24 × 8 = 312 units

R's contribution = 25 × 12 = 300 units

Total units = 256 + 312 + 300 = 868 units

Q's share of profit = (Q's units / Total units) × Total profit

⇒ Q's share = (312 / 868) × 28,210

⇒ Q's share = 10,140

∴ The correct answer is option 1.