MASINI Umet, rage No. 3, Q.No. 5
Q5. P and Q are sharing profits in the ratio of 5:3 admit R on 3/7th share in the new firm which
he takes 2/7th from P and 1/7th from Q. Calculate the
New Profit Sharing Ratio.
Answer:
For answer refer Unit-1, Page No. 4, Q.No. 8.
ESSAY QUESTIONS
(Nov/Des-19, 0200) | March/Ap-14 07



Answer :

To solve the problem of finding the new profit-sharing ratio when a new partner, R, is admitted and takes a certain fraction of the existing partners' shares, we will proceed step-by-step.

### Step-by-Step Solution:

1. Identify the original shares and contributions:
- P and Q are sharing profits in the ratio of 5:3.
- Let's assume the total profit is divided into 8 parts (5 parts for P and 3 parts for Q).

2. Determine the fractional shares:
- P's original share: [tex]\( \frac{5}{8} \)[/tex]
- Q's original share: [tex]\( \frac{3}{8} \)[/tex]

3. Calculate the fraction taken by R:
- R takes a total of [tex]\( \frac{3}{7} \)[/tex] of the new profit.
- This [tex]\( \frac{3}{7} \)[/tex] is made up of [tex]\( \frac{2}{7} \)[/tex] from P's share and [tex]\( \frac{1}{7} \)[/tex] from Q’s share.

4. Update P's and Q's shares after R takes his portion:
- P gives [tex]\( \frac{2}{7} \)[/tex] of his [tex]\( \frac{5}{8} \)[/tex] share to R:
[tex]\[ \text{P's new share} = \frac{5}{8} - \left( \frac{2}{7} \times \frac{5}{8} \right) \][/tex]
[tex]\[ \text{P's new share} = \frac{5}{8} - \frac{10}{56} = \frac{35}{56} - \frac{10}{56} = \frac{25}{56} \][/tex]

- Q gives [tex]\( \frac{1}{7} \)[/tex] of his [tex]\( \frac{3}{8} \)[/tex] share to R:
[tex]\[ \text{Q's new share} = \frac{3}{8} - \left( \frac{1}{7} \times \frac{3}{8} \right) \][/tex]
[tex]\[ \text{Q's new share} = \frac{3}{8} - \frac{3}{56} = \frac{21}{56} - \frac{3}{56} = \frac{18}{56} \][/tex]

5. Calculate R's share:
- R's share is the sum of what he takes from P and Q:
[tex]\[ \text{R's share} = \frac{2}{7} \times \frac{5}{8} + \frac{1}{7} \times \frac{3}{8} \][/tex]
[tex]\[ \text{R's share} = \frac{10}{56} + \frac{3}{56} = \frac{13}{56} \][/tex]

6. Verify the sum of all shares equals 1 (i.e., the whole profit):
[tex]\[ \text{Total shares} = \frac{25}{56} + \frac{18}{56} + \frac{13}{56} = \frac{56}{56} = 1 \][/tex]

7. Express the shares in the simplest ratio form:
The new shares of P, Q, and R are in the fraction form [tex]\( \frac{25}{56} : \frac{18}{56} : \frac{13}{56} \)[/tex].

8. Convert the fractions to a common denominator:
- Already common: [tex]\( \frac{25}{56} : \frac{18}{56} : \frac{13}{56} \)[/tex]
- The numerators represent their shares: [tex]\( 25 : 18 : 13 \)[/tex]

### Conclusion:
The new profit-sharing ratio for P, Q, and R is 25 : 18 : 13.