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The problem reads:

**"A, B, and C are partners sharing profits and losses in the ratio 5:3:2. As per the new agreement, A agreed to sacrifice ½ of his share in favor of B and C in the ratio of 3:2. Compute the new ratio and sacrifice and gain of the partners."**

### Step-by-step solution:

1. **Old Ratio:**
   The old profit-sharing ratio between A, B, and C is 5:3:2.

   - A's old share = \( \frac{5}{10} \)
   - B's old share = \( \frac{3}{10} \)
   - C's old share = \( \frac{2}{10} \)

2. **A's Sacrifice:**
   A agrees to sacrifice half of his share, i.e., \( \frac{1}{2} \times \frac{5}{10} = \frac{5}{20} \).

   This sacrificed share is divided between B and C in the ratio 3:2.
   - B's gain from A's sacrifice = \( \frac{3}{5} \times \frac{5}{20} = \frac{15}{100} = \frac{3}{20} \)
   - C's gain from A's sacrifice = \( \frac{2}{5} \times \frac{5}{20} = \frac{10}{100} = \frac{2}{20} \)

3. **New Ratio Calculation:**
   - A's new share = Old share of A - Sacrifice = \( \frac{5}{10} - \frac{5}{20} = \frac{10}{20} - \frac{5}{20} = \frac{5}{20} \)
   - B's new share = Old share of B + Gain = \( \frac{3}{10} + \frac{3}{20} = \frac{6}{20} + \frac{3}{20} = \frac{9}{20} \)
   - C's new share = Old share of C + Gain = \( \frac{2}{10} + \frac{2}{20} = \frac{4}{20} + \frac{2}{20} = \frac{6}{20} \)

4. **New Profit-sharing Ratio:**
   The new ratio of A, B, and C is \( 5:9:6 \).

5. **Sacrifice and Gain:**
   - A's sacrifice = \( \frac{5}{20} \)
   - B's gain = \( \frac{3}{20} \)
   - C's gain = \( \frac{2}{20} \)

### Final Answer:
- **New Ratio:** 5:9:6
- **Sacrifice of A:** \( \frac{5}{20} \)
- **Gain of B:** \( \frac{3}{20} \)
- **Gain of C:** \( \frac{2}{20} \)

Would you like further clarification on any part of the solution?

Here are some related questions:
1. How do you calculate profit-sharing ratios in partnerships?
2. What is the formula to calculate sacrifice ratio in partnerships?
3. How do you divide gains or sacrifices when given a specific ratio?
4. What are the implications of profit-sharing adjustments in real-life partnerships?
5. How can you represent a partner's gain or sacrifice as a percentage?

**Tip:** When redistributing shares, always ensure that the total percentage (or sum of ratios) still equals 100%.

A, B, and C are partners sharing profits and losses in the ratio 5:3:2. As per the new agreement, A agreed to sacrifice ½ of his share in favor of B and C in the ratio of 3:2. Compute new ratio and sacrifice and gain of the partners.

Solve the partnership problem where A sacrifices ½ of his share in favor of B and C in the ratio of 3:2. Learn how to compute the new profit-sharing ratio and the partners' gains and sacrifices with step-by-step solutions. Suitable for Grades 9-12.

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