Calculating the New Profit Sharing Ratio

This problem involves calculating the new profit sharing ratio for partners R, S, and the newly admitted partner T. When a new partner is admitted, the existing partners usually sacrifice a part of their share of profits to the new partner. The new profit sharing ratio is calculated based on the initial shares and the amount sacrificed by the old partners.

Initial Profit Sharing Ratio

Initially, R and S are partners sharing profits in the ratio of 5 ∶ 3. This means:

• R's initial share = $\frac{5}{5+3} = \frac{5}{8}$
• S's initial share = $\frac{3}{5+3} = \frac{3}{8}$

Calculating Partner Sacrifices

R gives 1/4th of his share, and S gives 2/5th of his share to the new partner T. It is important to note that the sacrifice is a fraction of their *own* share, not a fraction of the total profit.

• R's sacrifice = $\frac{1}{4}$ of R's share = $\frac{1}{4} \times \frac{5}{8} = \frac{5}{32}$
• S's sacrifice = $\frac{2}{5}$ of S's share = $\frac{2}{5} \times \frac{3}{8} = \frac{6}{40}$

These sacrifices represent the shares given up by R and S to T.

Calculating New Shares of R and S

The new share of each old partner will be their original share minus the amount they sacrificed.

• R's new share = R's initial share - R's sacrifice
• R's new share = $\frac{5}{8} - \frac{5}{32}$

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 8 and 32 is 32.

• R's new share = $\frac{5 \times 4}{8 \times 4} - \frac{5}{32} = \frac{20}{32} - \frac{5}{32} = \frac{15}{32}$

Now, for S's new share:

• S's new share = S's initial share - S's sacrifice
• S's new share = $\frac{3}{8} - \frac{6}{40}$

To subtract these fractions, we need a common denominator. The LCM of 8 and 40 is 40.

• S's new share = $\frac{3 \times 5}{8 \times 5} - \frac{6}{40} = \frac{15}{40} - \frac{6}{40} = \frac{9}{40}$

Calculating T's Share

The new partner T's share is the sum of the sacrifices made by R and S.

• T's share = R's sacrifice + S's sacrifice
• T's share = $\frac{5}{32} + \frac{6}{40}$

To add these fractions, we need a common denominator. The LCM of 32 and 40 is 160.

• T's share = $\frac{5 \times 5}{32 \times 5} + \frac{6 \times 4}{40 \times 4} = \frac{25}{160} + \frac{24}{160} = \frac{49}{160}$

Determining the New Profit Sharing Ratio

The new shares for R, S, and T are $\frac{15}{32}$, $\frac{9}{40}$, and $\frac{49}{160}$ respectively. To express this as a ratio, we need to find a common denominator for all three fractions. The LCM of 32, 40, and 160 is 160.

• R's new share (with denominator 160) = $\frac{15}{32} = \frac{15 \times 5}{32 \times 5} = \frac{75}{160}$
• S's new share (with denominator 160) = $\frac{9}{40} = \frac{9 \times 4}{40 \times 4} = \frac{36}{160}$
• T's share (with denominator 160) = $\frac{49}{160}$

The new profit sharing ratio of R, S, and T is the ratio of their new shares:

R : S : T = $\frac{75}{160} : \frac{36}{160} : \frac{49}{160}$

Since all fractions have the same denominator, the ratio is simply the ratio of the numerators:

R : S : T = 75 : 36 : 49

This is the new profit sharing ratio for R, S, and T after the admission of T. This process is fundamental in partnership accounting when dealing with the admission of partner and subsequent profit sharing ratio calculation based on share sacrifice.