Calculating Investment Ratios from Profit and Time

This problem involves understanding the relationship between investment, time period, and profit in a partnership or joint venture. The fundamental principle is that the profit earned by an individual is directly proportional to the product of their investment and the time period for which the investment was made.

The relationship can be expressed as:

Profit $\propto$ Investment $\times$ Time Period

Or, we can write it as:

Profit = Constant $\times$ Investment $\times$ Time Period

For comparing different individuals in the same venture, the 'Constant' is the same. Therefore, the ratio of profits is equal to the ratio of the products of their investments and time periods:

$\text{P}_A : \text{P}_B : \text{P}_C = (\text{I}_A \times \text{T}_A) : (\text{I}_B \times \text{T}_B) : (\text{I}_C \times \text{T}_C)$

where:

• $\text{P}_A, \text{P}_B, \text{P}_C$ are the profits of A, B, and C respectively.
• $\text{I}_A, \text{I}_B, \text{I}_C$ are the investments of A, B, and C respectively.
• $\text{T}_A, \text{T}_B, \text{T}_C$ are the time periods of investment for A, B, and C respectively.

The question provides the following ratios:

• Ratio of time periods ($\text{T}_A : \text{T}_B : \text{T}_C$) = 3 : 2 : 7
• Ratio of profits ($\text{P}_A : \text{P}_B : \text{P}_C$) = 4 : 3 : 14

We need to find the ratio of investments ($\text{I}_A : \text{I}_B : \text{I}_C$).

From the relationship Profit $\propto$ Investment $\times$ Time Period, we can deduce that Investment $\propto$ Profit / Time Period.

Therefore, the ratio of investments can be calculated as:

$\text{I}_A : \text{I}_B : \text{I}_C = \frac{\text{P}_A}{\text{T}_A} : \frac{\text{P}_B}{\text{T}_B} : \frac{\text{P}_C}{\text{T}_C}$

Substituting the given ratio values:

$\text{I}_A : \text{I}_B : \text{I}_C = \frac{4}{3} : \frac{3}{2} : \frac{14}{7}$

Simplify the fraction $\frac{14}{7}$:

$\frac{14}{7} = 2$

So the ratio becomes:

$\text{I}_A : \text{I}_B : \text{I}_C = \frac{4}{3} : \frac{3}{2} : 2$

To express this ratio in whole numbers, we need to find a common multiple of the denominators (3 and 2). The least common multiple (LCM) of 3 and 2 is 6.

Multiply each part of the ratio by 6:

$\text{I}_A : \text{I}_B : \text{I}_C = \left(\frac{4}{3} \times 6\right) : \left(\frac{3}{2} \times 6\right) : (2 \times 6)$

Calculate the values:

• $\frac{4}{3} \times 6 = 4 \times 2 = 8$
• $\frac{3}{2} \times 6 = 3 \times 3 = 9$
• $2 \times 6 = 12$

So, the ratio of investments is:

$\text{I}_A : \text{I}_B : \text{I}_C = 8 : 9 : 12$

Let's summarize the steps and results in a table:

The ratio of investments of A, B, and C is 8 : 9 : 12.

Revision Table: Profit, Investment, and Time Ratios

Understanding the relationship between these three factors is key to solving partnership problems. This table helps recap the formulas:

Additional Information on Partnership Ratios

In partnership problems, profits are typically shared based on the capital invested and the duration of the investment. If investments are for different time periods, the equivalent capital for a standard time unit (e.g., one month or one year) is calculated by multiplying the investment amount by the time period. The profits are then divided in the ratio of these equivalent capitals.

For example, if A invests $\text{I}_A$ for $\text{T}_A$ months and B invests $\text{I}_B$ for $\text{T}_B$ months, their profit sharing ratio will be $(\text{I}_A \times \text{T}_A) : (\text{I}_B \times \text{T}_B)$. This is exactly what was used in reverse in the problem above to find the investment ratio when profit and time ratios were given.

It is important to ensure that the units of time are consistent for all partners (e.g., all in months or all in years).