A and B started business together by investing Rs. 45,000 in ratio of 2 : 3. If after 3 months C invests Rs. 30,000 then what would be the ratio of their profit sharing at end of one year (A : B : C)?
4 : 6 : 5
Understanding Partnership Profit Sharing
This problem involves calculating the profit-sharing ratio among partners in a business. The profit share depends on both the amount invested and the duration for which the investment was made. We need to determine the individual investments of partners A and B, and then factor in C's investment and the time duration each partner was in the business to find the final profit-sharing ratio at the end of one year.
Calculating Individual Investments
The total investment made by A and B is Rs. 45,000, and they invested in the ratio of 2 : 3.
- Total ratio parts = 2 + 3 = 5
- A's investment = $( \frac{2}{5} ) \times 45,000 = 2 \times 9,000 = \text{Rs. } 18,000$
- B's investment = $( \frac{3}{5} ) \times 45,000 = 3 \times 9,000 = \text{Rs. } 27,000$
Calculating Profit Share Factors
Profit sharing is calculated based on the product of the investment amount and the duration (in months) for which the investment was active. The total duration is one year, which is 12 months.
- Partner A: Invested Rs. 18,000 for the entire 12 months. A's profit share factor = $18,000 \times 12 = 216,000$
- Partner B: Invested Rs. 27,000 for the entire 12 months. B's profit share factor = $27,000 \times 12 = 324,000$
- Partner C: Invested Rs. 30,000 after 3 months. This means C's investment was active for $12 - 3 = 9$ months. C's profit share factor = $30,000 \times 9 = 270,000$
Determining the Profit Sharing Ratio
The ratio of their profits will be the ratio of their respective profit share factors.
Profit Sharing Ratio (A : B : C) = A's factor : B's factor : C's factor
Ratio = $216,000 : 324,000 : 270,000$
To simplify the ratio, we can divide all parts by their common factors. First, divide by 1,000:
Ratio = $216 : 324 : 270$
Now, we find the greatest common divisor (GCD) for these numbers. Let's divide by common factors:
- Divide by 2: $108 : 162 : 135$
- Divide by 3: $36 : 54 : 45$
- Divide by 9: $4 : 6 : 5$
The simplified profit-sharing ratio is 4 : 6 : 5.
Final Profit Sharing Ratio
The profit sharing ratio at the end of one year for A, B, and C is 4 : 6 : 5.
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