Three partners shared the profit in a business in the proportion of 9 ∶ 8 ∶ 11. They invested their capitals for 4 months, 6 months and 18 months, respectively. What was the ratio of their capitals?
81 ∶ 48 ∶ 22
Understanding Profit Sharing and Capital Investment
In a business partnership, the profit earned is typically shared among the partners based on their investment. The share of profit for each partner is directly proportional to the capital they invest and the duration for which the capital is invested. This means:
$$ \text{Profit} \propto \text{Capital} \times \text{Time} $$
If three partners invest capitals \(C_1, C_2, C_3\) for times \(T_1, T_2, T_3\) respectively, and their profits are \(P_1, P_2, P_3\), then the ratio of their profits is given by:
$$ P_1 : P_2 : P_3 = (C_1 \times T_1) : (C_2 \times T_2) : (C_3 \times T_3) $$
From this relationship, we can also find the ratio of their capitals if we know the profit ratio and the time periods of investment. Rearranging the formula, we get:
$$ C \propto \frac{\text{Profit}}{\text{Time}} $$
Therefore, the ratio of their capitals is:
$$ C_1 : C_2 : C_3 = \frac{P_1}{T_1} : \frac{P_2}{T_2} : \frac{P_3}{T_3} $$
Calculating the Ratio of Capitals
The problem provides the following information about three partners:
- Ratio of their profits \(P_1 : P_2 : P_3 = 9 : 8 : 11\).
- Time for which they invested their capitals: \(T_1 = 4\) months, \(T_2 = 6\) months, \(T_3 = 18\) months.
We need to find the ratio of their capitals \(C_1 : C_2 : C_3\). Using the relationship derived above, we have:
$$ C_1 : C_2 : C_3 = \frac{9}{4} : \frac{8}{6} : \frac{11}{18} $$
To simplify this ratio involving fractions, we need to find a common denominator for the denominators 4, 6, and 18. The least common multiple (LCM) of 4, 6, and 18 is 36.
Now, multiply each fraction in the ratio by the LCM (36) to clear the denominators:
$$ C_1 : C_2 : C_3 = \left(\frac{9}{4} \times 36\right) : \left(\frac{8}{6} \times 36\right) : \left(\frac{11}{18} \times 36\right) $$
Perform the multiplication and simplification:
- First partner: \(\frac{9}{4} \times 36 = 9 \times \frac{36}{4} = 9 \times 9 = 81\)
- Second partner: \(\frac{8}{6} \times 36 = 8 \times \frac{36}{6} = 8 \times 6 = 48\)
- Third partner: \(\frac{11}{18} \times 36 = 11 \times \frac{36}{18} = 11 \times 2 = 22\)
So, the ratio of their capitals is \(81 : 48 : 22\).
Summary of Calculation
| Partner | Profit Ratio Share (P) | Time (T in months) | Capital Ratio Share (P/T) | Capital Ratio (P/T * LCM) |
|---|---|---|---|---|
| 1 | 9 | 4 | $$\frac{9}{4}$$ | $$\frac{9}{4} \times 36 = 81$$ |
| 2 | 8 | 6 | $$\frac{8}{6}$$ | $$\frac{8}{6} \times 36 = 48$$ |
| 3 | 11 | 18 | $$\frac{11}{18}$$ | $$\frac{11}{18} \times 36 = 22$$ |
The resulting ratio of the capitals is \(81 : 48 : 22\).
Revision Table: Partnership Calculations
| Concept | Formula | Explanation |
|---|---|---|
| Profit Ratio | $$P_1:P_2:P_3 = (C_1 T_1) : (C_2 T_2) : (C_3 T_3)$$ | Profit is directly proportional to Capital and Time. |
| Capital Ratio (given Profit & Time) | $$C_1:C_2:C_3 = \frac{P_1}{T_1} : \frac{P_2}{T_2} : \frac{P_3}{T_3}$$ | Capital is directly proportional to Profit and inversely proportional to Time. |
| Time Ratio (given Profit & Capital) | $$T_1:T_2:T_3 = \frac{P_1}{C_1} : \frac{P_2}{C_2} : \frac{P_3}{C_3}$$ | Time is directly proportional to Profit and inversely proportional to Capital. |
Additional Information on Partnership Business
A partnership is a type of business organization where two or more individuals agree to share the profits or losses of a business that is carried on by all or any of them acting for all. Key aspects include:
- Agreement: Partnerships are formed through an agreement, which can be oral or written. A written agreement (Partnership Deed) is highly recommended to avoid disputes.
- Sharing of Profits and Losses: Partners share profits and losses in the ratio agreed upon in the partnership deed. If there is no agreement, profits and losses are shared equally.
- Liability: Generally, partners have unlimited liability, meaning their personal assets can be used to pay off business debts.
- Mutual Agency: Each partner is an agent for all other partners and the firm. The act of one partner binds the firm and other partners.
Understanding the relationship between capital, time, and profit is fundamental in solving problems related to partnership accounts and profit distribution.
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