Profit Ratio Calculation

The total profit is Rs. 33,000. E's share is Rs. 15,000. Therefore, D's share is:

D's Profit = Total Profit - E's Profit

D's Profit = Rs. 33,000 - Rs. 15,000 = Rs. 18,000

The ratio of their profits is D:E = 18,000 : 15,000. Simplifying this ratio by dividing both by 3,000 gives:

Profit Ratio (D:E) = 6 : 5

Investment Calculation

Let the initial investment made by both D and E be Rs. x.

The business duration is 1 year (12 months).

• E's Investment: E invested Rs. x for the entire 12 months. Total investment by E = $x \times 12 = 12x$ months.
• D's Investment: D invested Rs. x for the first 4 months. After 4 months, D added Rs. 10,000. So, for the remaining 8 months (12 - 4 = 8), D invested Rs. $(x + 10,000)$. Total investment by D = $(x \times 4) + ((x + 10,000) \times 8)$ = $4x + 8x + 80,000$ = $12x + 80,000$ months.

Equating Ratios and Finding Initial Investment

The ratio of total investments must be equal to the ratio of profits.

Investment Ratio (D:E) = Profit Ratio (D:E)

$\frac{12x + 80,000}{12x} = \frac{6}{5}$

Cross-multiply to solve for x:

$5 \times (12x + 80,000) = 6 \times 12x$

$60x + 400,000 = 72x$

Subtract $60x$ from both sides:

$400,000 = 72x - 60x$

$400,000 = 12x$

Divide by 12:

$x = \frac{400,000}{12}$

$x = \frac{100,000}{3}$

$x \approx 33,333.33$

The initial investment was approximately Rs. 33,333.33.