To solve the problem, let's walk through the steps to determine the new profit sharing ratio of [tex]\( Q, R, \)[/tex] and [tex]\( S \)[/tex] after [tex]\( P \)[/tex] retires and his share is distributed between [tex]\( R \)[/tex] and [tex]\( S \)[/tex].
1. Initial Profit Sharing Ratio:
[tex]\[ P:Q:R:S = 4:3:2:1 \][/tex]
2. Share of each partner:
The total ratio parts are [tex]\( 4 + 3 + 2 + 1 = 10 \)[/tex].
- [tex]\( P \)[/tex] share is [tex]\(\frac{4}{10}\)[/tex]
- [tex]\( Q \)[/tex] share is [tex]\(\frac{3}{10}\)[/tex]
- [tex]\( R \)[/tex] share is [tex]\(\frac{2}{10}\)[/tex]
- [tex]\( S \)[/tex] share is [tex]\(\frac{1}{10}\)[/tex]
3. Distribution of [tex]\( P \)[/tex]'s share:
[tex]\( P )'s share of \(\frac{4}{10}\)[/tex] is taken over by [tex]\( R \)[/tex] and [tex]\( S \)[/tex] in the ratio 1:2.
- The total parts for [tex]\( R \)[/tex] and [tex]\( S \)[/tex] to take are [tex]\( 1+2=3 \)[/tex].
- [tex]\( R )'s additional share will be \(\frac{1}{3} \)[/tex] of [tex]\(\frac{4}{10},\)[/tex] which equals [tex]\(\frac{4}{30} \)[/tex].
- [tex]\( S )'s additional share will be \(\frac{2}{3} \)[/tex] of [tex]\(\frac{4}{10},\)[/tex] which equals [tex]\(\frac{8}{30} \)[/tex].
4. New Shares of [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:
Adding the new shares to [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:
- [tex]\( R )'s new share is \(\frac{2}{10} + \(\frac{4}{30} = \(\frac{2}{10} + \(\frac{2}{15} = \(\frac{6}{30} + \(\frac{2}{15} = \(\frac{10}{30} .
- \( S )'s new share is \(\frac{1}{10} + \ ( \(\frac{8}{30} = \(\frac{3}{30} + \(\frac{8}{30} = \(\frac{9}{30} \)[/tex].
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5. Resulting shares in common ratio:
Since Q's share remains unaffected, we find its part as follows:
- [tex]\( Q )'s share remains \(\frac{3}{10} = \(\frac{9}{30} \)[/tex].
Total shares [tex]\(\frac{9}{30}+ \(\frac{10}{30}+ \(\frac{9}{30}= \(\frac{28}{30}
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6. Final Simplified Ratios:
- \( Q's final share representing= \ \frac{9}{28}\)[/tex]
[tex]\( R's final share representing= \ \frac{10 }{28}\}
\( S \ = final share representing= \ \frac{9 }(\28 \)[/tex]
Ratios ==9:10:\9:
Therefore, the new profit-sharing ratio of [tex]\( Q, R, \)[/tex] and [tex]\( S \)[/tex] after [tex]\( P \)[/tex]'s retirement is [tex]\( 3: 3: 3 \)[/tex].
