Understanding the Profit Sharing Problem

The question asks us to determine the share of B when a total profit of Rs. 29,120 is divided among A, B, and C based on specific relationships between their shares. We are given that A's share is $2\frac{1}{2}$ times B's share, and B's share is $1\frac{1}{5}$ times C's share.

Setting Up the Relationships Between Shares

First, let's write down the given relationships mathematically:

• A's share is $2\frac{1}{2}$ times B's share. Converting the mixed number to an improper fraction, $2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$. So, A's share = $\frac{5}{2} \times$ B's share.
• B's share is $1\frac{1}{5}$ times C's share. Converting the mixed number to an improper fraction, $1\frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5}$. So, B's share = $\frac{6}{5} \times$ C's share.

To find the ratio of shares A : B : C, it's easiest to express all shares in terms of a single variable, for example, C's share.

• We know B's share = $\frac{6}{5} \times$ C's share.
• Now, substitute the expression for B's share into the equation for A's share:
  A's share = $\frac{5}{2} \times$ (B's share)
  A's share = $\frac{5}{2} \times \left(\frac{6}{5} \times \text{ C's share}\right)$
  A's share = $\frac{5 \times 6}{2 \times 5} \times$ C's share
  A's share = $\frac{30}{10} \times$ C's share
  A's share = $3 \times$ C's share.

Determining the Ratio A : B : C

Based on our calculations, we have the shares in terms of C:

• A's share = $3 \times$ C's share
• B's share = $\frac{6}{5} \times$ C's share
• C's share = $1 \times$ C's share

So, the ratio A : B : C is $3 : \frac{6}{5} : 1$.

To get a simple ratio with whole numbers, we need to multiply each part of the ratio by the least common multiple (LCM) of the denominators, which is 5.

Ratio A : B : C = $\left(3 \times 5\right) : \left(\frac{6}{5} \times 5\right) : \left(1 \times 5\right)$

Ratio A : B : C = $15 : 6 : 5$.

Calculating B's Share

The total profit is Rs. 29,120. This total profit is divided according to the ratio 15 : 6 : 5 for A : B : C.

The total number of ratio parts is the sum of the individual parts:

Total parts = $15 + 6 + 5 = 26$ parts.

Each part of the ratio represents an equal share of the total profit. To find the value of one ratio part, we divide the total profit by the total number of parts:

Value of one part = $\frac{\text{Total Profit}}{\text{Total Ratio Parts}}$

Value of one part = $\frac{29,120}{26}$

Let's perform the division:

\(\frac{29120}{26} = \frac{14560}{13}\)

\(\frac{14560}{13} = 1120\)

So, the value of one ratio part is Rs. 1,120.

Now we can find B's share. B's share corresponds to 6 parts in the ratio 15 : 6 : 5.

B's share = Number of parts for B $\times$ Value of one part

B's share = $6 \times 1,120$

B's share = Rs. 6,720.

Let's quickly verify the shares of A and C:

• A's share = 15 parts $\times$ Rs. 1,120/part = Rs. 16,800.
• C's share = 5 parts $\times$ Rs. 1,120/part = Rs. 5,600.
• Total profit = A's share + B's share + C's share = Rs. 16,800 + Rs. 6,720 + Rs. 5,600 = Rs. 29,120.

The sum of the calculated shares equals the total given profit, confirming our calculations are correct.

Summary of Shares

Therefore, B's share is Rs. 6,720.

Revision Table: Profit Sharing Ratios

Additional Information: Profit and Loss Sharing Basics

Profit and loss sharing is a common concept in partnerships and business where earnings or losses are distributed among partners based on agreed-upon terms, often represented as ratios. Understanding how to work with ratios is crucial for solving such problems.

• Ratios: A ratio compares two or more quantities. For instance, a ratio 2:3 means the first quantity is two parts for every three parts of the second.
• Proportionality: In profit sharing, each individual's share is proportional to their respective part in the ratio. If the total profit increases or decreases, each person's share changes proportionally.
• Steps to Solve Ratio Problems:
  • Understand the relationships given and write them mathematically.
  • Express all quantities in terms of a common variable or find a combined ratio.
  • Calculate the total number of ratio parts.
  • Find the value of one ratio part by dividing the total amount by the total parts.
  • Calculate each individual's share by multiplying their ratio part by the value of one part.

These steps are generally applicable to various problems involving the division of a quantity according to a given ratio, whether it's profit, loss, investment, or any other measurable quantity.