Question

P and Q are partners sharing profits in the ratio of 5:3. They admit R for 1/4th share in profits, which he acquires equally from P and Q.
The sacrificing ratio of P and Q will be:

• A. 5 : 3
• B. 1 : 1
• C. 3 : 2
• D. 7 : 5

Hint:

If the question explicitly states that the new partner acquires their share "equally" from the existing partners, you do not need to perform any mathematical calculations to find the sacrificing ratio.
The word "equally" directly implies a sacrificing ratio of 1:1.
Always read the acquisition terms carefully to spot these keywords immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
In this problem, we need to find the sacrificing ratio of partners P and Q when a new partner, R, is admitted.
The old profit-sharing ratio between P and Q is 5:3.
The new partner, R, is admitted for a 1/4th share of the profits.
The key piece of information is that R acquires his 1/4th share equally from P and Q.
We need to determine the ratio in which P and Q give up their profit shares to accommodate R.

Step 2: Key Formula or Approach:
1. Compute the individual sacrifice made by each of the existing partners:
\[ \text{Sacrifice of an Old Partner} = \text{Share acquired by New Partner} \times \text{Acquisition Proportion} \]
2. Compare the calculated individual sacrifices of P and Q to determine their sacrificing ratio:
\[ \text{Sacrificing Ratio} = \text{Sacrifice of P} : \text{Sacrifice of Q} \]

Step 3: Detailed Explanation:
1. R is admitted for a 1/4th share in the profits of the firm.
2. The problem states that R acquires this 1/4th share equally from both P and Q.
This means both P and Q contribute an equal proportion of R's incoming share.
3. Let us calculate the numerical value of the sacrifice made by P:
\[ \text{Sacrifice of P} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]
4. Next, we calculate the numerical value of the sacrifice made by Q:
\[ \text{Sacrifice of Q} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]
5. Since both P and Q are sacrificing exactly \(\frac{1}{8}\) of their profit shares, we compare their sacrifices:
\[ \text{Sacrificing Ratio} = \frac{1}{8} : \frac{1}{8} \]
Simplifying this ratio gives:
\[ \text{Sacrificing Ratio} = 1 : 1 \]
6. To verify this, let us calculate the new profit-sharing ratio:
\[ \text{P's New Share} = \text{Old Share} - \text{Sacrifice} = \frac{5}{8} - \frac{1}{8} = \frac{4}{8} \]
\[ \text{Q's New Share} = \text{Old Share} - \text{Sacrifice} = \frac{3}{8} - \frac{1}{8} = \frac{2}{8} \]
\[ \text{R's Share} = \frac{1}{4} = \frac{2}{8} \]
The new profit-sharing ratio of P, Q, and R is \(4:2:2\), which simplifies to \(2:1:1\).
This confirms that the proportion of profits surrendered by P and Q is indeed equal, which is \(1:1\).

Step 4: Final Answer:
The sacrificing ratio of P and Q is 1:1.
Thus, Option (B) is the correct answer.