Rakesh, Amit, and Suresh started a business. Rakesh invested 1/2 part, Amit 1/3 part, and the rest of the capital was invested by Suresh. The ratio of their profits will be:
2:3:1
Solving the Business Profit Ratio Problem
In this business partnership problem, three individuals, Rakesh, Amit, and Suresh, invest different parts of the total capital. The profit in a partnership is typically shared in the ratio of the capital invested by each partner, assuming the investment period is the same for everyone (which is implied here as no time is mentioned).
Calculating Individual Investments
We are given the investment parts for Rakesh and Amit as fractions of the total capital. Let the total capital be represented by 1 (or any common unit).
- Rakesh's investment part = $\frac{1}{2}$
- Amit's investment part = $\frac{1}{3}$
The remaining capital is invested by Suresh. To find Suresh's part, we first calculate the total part invested by Rakesh and Amit:
Combined investment of Rakesh and Amit = Rakesh's part + Amit's part
Combined investment = $\frac{1}{2} + \frac{1}{3}$
To add these fractions, we find a common denominator, which is the least common multiple (LCM) of 2 and 3. The LCM of 2 and 3 is 6.
Combined investment = $\frac{1 \times 3}{2 \times 3} + \frac{1 \times 2}{3 \times 2} = \frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$
So, Rakesh and Amit together invested $\frac{5}{6}$ of the total capital.
Suresh's investment part is the rest of the capital. Assuming the total capital is 1, Suresh's part is:
Suresh's investment part = Total capital - Combined investment of Rakesh and Amit
Suresh's investment part = $1 - \frac{5}{6}$
To subtract, we write 1 as $\frac{6}{6}$:
Suresh's investment part = $\frac{6}{6} - \frac{5}{6} = \frac{6-5}{6} = \frac{1}{6}$
So, Suresh invested $\frac{1}{6}$ of the total capital.
Determining the Investment Ratio
Now we have the investment parts for all three partners:
- Rakesh: $\frac{1}{2}$
- Amit: $\frac{1}{3}$
- Suresh: $\frac{1}{6}$
The ratio of their investments (Rakesh : Amit : Suresh) is $\frac{1}{2} : \frac{1}{3} : \frac{1}{6}$.
To express this ratio in whole numbers, we multiply each fraction by the LCM of the denominators (2, 3, and 6). The LCM of 2, 3, and 6 is 6.
- Rakesh's ratio part = $\frac{1}{2} \times 6 = 3$
- Amit's ratio part = $\frac{1}{3} \times 6 = 2$
- Suresh's ratio part = $\frac{1}{6} \times 6 = 1$
So, the ratio of their investments is 3 : 2 : 1.
| Partner | Investment Part (Fraction) | Investment Part (Whole Number Ratio) |
|---|---|---|
| Rakesh | $\frac{1}{2}$ | $\frac{1}{2} \times 6 = 3$ |
| Amit | $\frac{1}{3}$ | $\frac{1}{3} \times 6 = 2$ |
| Suresh | $\frac{1}{6}$ | $\frac{1}{6} \times 6 = 1$ |
Profit Sharing Ratio
In a business partnership, if the investments are for the same duration, the profits are shared in the same ratio as the investments.
Therefore, the ratio of their profits (Rakesh : Amit : Suresh) will be the same as the ratio of their investments.
Profit Ratio = Investment Ratio = 3 : 2 : 1
The ratio of their profits is 3:2:1.
Revision Table: Business Profit Ratio Calculation
| Concept | Details |
|---|---|
| Rakesh's Investment | $\frac{1}{2}$ of total capital |
| Amit's Investment | $\frac{1}{3}$ of total capital |
| Combined Rakesh & Amit Investment | $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ |
| Suresh's Investment | $1 - \frac{5}{6} = \frac{1}{6}$ |
| Investment Ratio (Rakesh : Amit : Suresh) | $\frac{1}{2} : \frac{1}{3} : \frac{1}{6}$ |
| Convert to Whole Number Ratio | Multiply by LCM(2, 3, 6) = 6 |
| Whole Number Investment Ratio | $3 : 2 : 1$ |
| Profit Ratio | Same as Investment Ratio ($3 : 2 : 1$) |
Additional Information on Business Partnerships and Ratios
Understanding ratios in business partnerships is fundamental. The profit distribution method depends on the partnership agreement, but often it's based on the capital contribution ratio.
- Capital Contribution: This is the amount of money or assets each partner invests in the business.
- Profit and Loss Sharing Ratio: This is the agreed ratio in which profits and losses are divided among partners. If there is no agreement, the law usually dictates equal sharing, but when capital ratios are given, profit is often proportional to capital.
- Time Factor: If partners invest capital for different durations, the profit sharing ratio is usually calculated based on the product of capital and time (Capital $\times$ Time). For example, if Rakesh invests $\text{C}_R$ for $\text{T}_R$ months and Amit invests $\text{C}_A$ for $\text{T}_A$ months, their profit ratio would be $(\text{C}_R \times \text{T}_R) : (\text{C}_A \times \text{T}_A)$. In this question, the time period is assumed to be the same for all, so the ratio is simply based on capital.
- Converting Fractional Ratios: To convert a ratio like $\frac{a}{x} : \frac{b}{y} : \frac{c}{z}$ to a whole number ratio, find the LCM of the denominators ($x, y, z$). Then multiply each fraction by the LCM. For example, $\frac{1}{2} : \frac{1}{3} : \frac{1}{6}$ with LCM=6 becomes $(\frac{1}{2} \times 6) : (\frac{1}{3} \times 6) : (\frac{1}{6} \times 6) = 3 : 2 : 1$.
Important Questions from Non-Verbal Reasoning
Choose the alternative which most closely resembles the water image of the given figures.
Select a figure from the given four alternatives, which when placed in the blank space of the problem figure would complete the pattern.
Choose the alternative which most closely resembles the mirror image of the given combination: MARKER
Among the four answer figures, which one can be formed from the cut-out pieces given below?
Find the alternative which contains figure (A) as its part.
