Question

Hema and Tara were partners in a firm sharing profits and losses in the ratio of \( 2 : 3 \). They admitted Ojas as a new partner. Hema surrendered \( \frac{1}{3} \) of her share and Tara surrendered \( \frac{1}{2} \) of her share in favour of Ojas. The new profit-sharing ratio of Hema, Tara and Ojas will be:

• A. 8 : 9 : 13
• B. 3 : 2 : 5
• C. 2 : 3 : 5
• D. 2 : 3 : 25

Hint:

To calculate the new profit-sharing ratio, subtract the surrendered shares from the original shares and add them to the new partner's share.

Answer 1

The Correct Option is A

1. Initial Allocation: Hema's initial proportion was \( \frac{2}{5} \), and Tara's was \( \frac{3}{5} \). 2. Portions Relinquished: Hema relinquished \( \frac{1}{3} \) of her allocation: \[ \text{Hema's relinquished portion} = \frac{1}{3} \times \frac{2}{5} = \frac{2}{15}. \] Tara relinquished \( \frac{1}{2} \) of her allocation: \[ \text{Tara's relinquished portion} = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10}. \] 3. Ojas's Entitlement: Ojas's total entitlement is the sum of Hema's and Tara's relinquished portions: \[ \text{Ojas's entitlement} = \frac{2}{15} + \frac{3}{10} = \frac{4}{30} + \frac{9}{30} = \frac{13}{30}. \] 4. Remaining Allocations: Hema's revised allocation = \( \frac{2}{5} - \frac{2}{15} = \frac{6}{15} - \frac{2}{15} = \frac{4}{15}. \) Tara's revised allocation = \( \frac{3}{5} - \frac{3}{10} = \frac{6}{10} - \frac{3}{10} = \frac{3}{10}. \) 5. Revised Ratio: To establish the new ratio, all allocations are converted to a common denominator (LCM = 30): \[ \text{Hema's allocation} = \frac{4}{15} = \frac{8}{30}, \quad \text{Tara's allocation} = \frac{3}{10} = \frac{9}{30}, \quad \text{Ojas's allocation} = \frac{13}{30}. \] The resulting new ratio is \( 8 : 9 : 13 \).