A,B,C,D share profits in the ratio 6:5:12:3, B left the business for doing MBA, what is the new ratio between A,C,D, divide B’s ratio amongst rest 3 

When B leaves the business, and the remaining partners A, C, and D take over B's share of the profits, B's share will be distributed among A, C, and D in proportion to their existing shares.

Here's how to calculate the new ratio:

1. **Original Ratios:**
   - A : B : C : D = 6 : 5 : 12 : 3

2. **Distribute B's Share:**
   - The sum of the ratios of A, C, and D is \(6 + 12 + 3 = 21\).
   - B's share (5 parts) will be divided among A, C, and D in the ratio of their original shares \(6:12:3\).

   For each partner:
   - A's new share = \(6 + \frac{6}{21} \times 5\)
   - C's new share = \(12 + \frac{12}{21} \times 5\)
   - D's new share = \(3 + \frac{3}{21} \times 5\)

3. **Calculations:**

   - A's new share = \(6 + \frac{6}{21} \times 5 = 6 + \frac{30}{21} = 6 + \frac{10}{7} = \frac{42}{7} + \frac{10}{7} = \frac{52}{7}\)
   - C's new share = \(12 + \frac{12}{21} \times 5 = 12 + \frac{60}{21} = 12 + \frac{20}{7} = \frac{84}{7} + \frac{20}{7} = \frac{104}{7}\)
   - D's new share = \(3 + \frac{3}{21} \times 5 = 3 + \frac{15}{21} = 3 + \frac{5}{7} = \frac{21}{7} + \frac{5}{7} = \frac{26}{7}\)

4. **Simplified New Ratio:**

   \[
   A : C : D = \frac{52}{7} : \frac{104}{7} : \frac{26}{7}
   \]
   Simplifying, the ratio becomes \(52:104:26\).

5. **Final Simplified Ratio:**

   - Dividing each term by the GCD (26):

   \[
   A : C : D = \frac{52}{26} : \frac{104}{26} : \frac{26}{26} = 2:4:1
   \]

So, the new ratio between A, C, and D after redistributing B's share is **2:4:1**.

Do you have any questions or need further details? Here are some related questions:

1. How would the ratio change if B's share was divided differently among A, C, and D?
2. What if B returns and contributes twice as much as before?
3. How would the overall profit affect the individual shares?
4. Can this method be used if multiple partners leave?
5. What if B's share was divided unequally among A, C, and D?

**Tip:** Always ensure to check the sum of new ratios to confirm the distribution is correct.

Explore how to redistribute B's share in a business where A, B, C, and D share profits in the ratio 6:5:12:3. Learn the new ratio between A, C, and D after B leaves for MBA, divided proportionally among them. Detailed step-by-step solution included.

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