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.
