Understanding the Partnership Investments and Timeline

This problem involves calculating the total profit of a business based on the investments and duration each partner stayed in the business. We need to find the share of each partner and then determine the total profit using the known profit share of one partner (C).

Let's break down the information given:

• Partner A's Investment: ₹ $60,000$ for the full year (12 months).
• Partner B's Investment: ₹ $90,000$ for the first 5 months.
• Partner C's Joining: C joined after B left (i.e., after 5 months).
• Partner C's Investment: ₹ $60,000$ less than B's investment.
• C's Capital Calculation: ₹ $90,000$ (B's capital) - ₹ $60,000$ = ₹ $30,000$.
• Partner C's Duration: Since B left after 5 months, C was in the business for the remaining $12 - 5 = 7$ months.
• C's Profit Share: ₹ $42,000$ at the end of the year.

Calculating Profit Ratio Based on Investment and Time

The profit shared among partners is usually in proportion to the product of their investment and the duration for which they invested (Investment $\times$ Time).

• A's Profit Contribution: ₹ $60,000 \times 12$ months = ₹ $7,20,000$
• B's Profit Contribution: ₹ $90,000 \times 5$ months = ₹ $4,50,000$
• C's Profit Contribution: ₹ $30,000 \times 7$ months = ₹ $2,10,000$

The ratio of their profit contributions (A : B : C) is: $ 7,20,000 : 4,50,000 : 2,10,000 $

To simplify the ratio, we can divide all parts by common factors. Let's divide by $10,000$: $ 72 : 45 : 21 $ Now, let's divide by $3$: $ 24 : 15 : 7 $ So, the ratio of profit sharing is $24 : 15 : 7$.

Determining Total Profit from C's Share

We know that C's profit share is ₹ $42,000$, and this corresponds to 7 parts of the total profit ratio ($24 + 15 + 7 = 46$ parts).

This means that C's 7 parts represent ₹ $42,000$. We can set up a proportion to find the total profit: $ \frac{\text{C's Profit Share}}{\text{C's Ratio Part}} = \frac{\text{Total Profit}}{\text{Total Ratio Parts}} $ $ \frac{₹ 42,000}{7} = \frac{\text{Total Profit}}{46} $

Now, we can solve for the Total Profit: $ \text{Total Profit} = \frac{₹ 42,000}{7} \times 46 $ $ \text{Total Profit} = ₹ 6,000 \times 46 $ $ \text{Total Profit} = ₹ 2,76,000 $

Therefore, the total profit earned at the end of the year was ₹ $2,76,000$.