Question

Q. 19. A and B are partners in a firm sharing profits in the ratio of 3 : 1. They admit
C and decide that the profit-sharing ratio between B and C shall be same as existing
between A and B. Calculate new profit-sharing ratio and the sacrificing ratio.
[Ans. New Ratio 9:3:1; Sacrificing Ratio = 3 : 1.]​

Answer 1

ANSWER :

• ❖ If A and B are partners in a firm sharing profits in the ratio of 3 : 1 and they admit C and decide that the profit-sharing ratio between B and C shall be same as existing between A and B; then the New Profit Sharing Ratio among A, B and C is 9 : 3 : 1 and the Sacrificing Ratio between A and B is 3 : 1.

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SOLUTION :

❒ Given :-

• A and B are partners in a firm sharing profits in the ratio of 3 : 1.

• After admitting C into the firm, the profit-sharing ratio between B and C shall be same as existing between A and B.

❒ To Calculate :-

• New Profit Sharing Ratio among A, B and C = ?

• Sacrificing Ratio between A and B = ?

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❒ Calculation of New Profit Sharing Ratio :-

It is given that,

• A and B are partners in a firm sharing profits in the ratio of 3 : 1.

Hence,

• A : B = 3 : 1

And,

• After admitting C into the firm, the profit-sharing ratio between B and C shall be same as existing between A and B.

Hence,

• B : C = 3 : 1

Thus,

• $\bigstar \: \: \sf{ \dfrac{A}{B} = \dfrac{3}{1}}$

$\implies\: \: \sf{ \dfrac{A}{B} = \dfrac{3 \times 3}{1 \times 3}}$

$\implies\: \: \sf{ \dfrac{A}{B} = \dfrac{9}{ 3}} \: \: \: \: \:$

$\: \therefore \: \: \sf{A : B = 9 : 3 \: }$ ----------> (1)

And,

• $\bigstar \: \: \sf{ \dfrac{B}{C} = \dfrac{3}{1}}$

$\implies\: \: \sf{ \dfrac{B}{C} = \dfrac{3 \times 1}{1 \times 1}}$

$\implies\: \: \sf{ \dfrac{B}{C} = \dfrac{3}{1}} \: \: \: \: \:$

$\: \therefore \: \: \sf{B : C = 3 : 1 \: }$ ----------> (2)

From Eq. (1) and Eq. (2), we get,

• ✪ A : B : C = 9 : 3 : 1

∴ New Profit Sharing Ratio among A, B and C is 9 : 3 : 1.

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❒ Calculation of Sacrificing Ratio :-

We have,

• Old Profit Sharing Ratio between A and B = 3 : 1

So,

• ➺ A's Old Share = $\sf{\dfrac{3}{4}}$

• ➺ B's Old Share = $\sf{\dfrac{1}{4}}$

And,

• New Profit Sharing Ratio between A, B and C = 9 : 3 : 1

So,

• ➺ A's New Share = $\sf{\dfrac{9}{13}}$

• ➺ B's New Share = $\sf{\dfrac{3}{13}}$

We know that,

• $\dag \: \: \underline{ \boxed{ \sf{ \: Sacrifice = Old \: \: Share - New \: \: Share \: }}}$

Using this formula, we get,

• ★ A's Sacrifice = A's Old Share - A's New Share

➜ A's Sacrifice = $\sf{\dfrac{3}{4}}$ - $\sf{\dfrac{9}{13}}$

➜ A's Sacrifice = $\sf{\dfrac{39 - 36}{52}}$

➜ A's Sacrifice = $\sf{\dfrac{3}{52}}$

And,

• ★ B's Sacrifice = B's Old Share - B's New Share

➜ B's Sacrifice = $\sf{\dfrac{1}{4}}$ - $\sf{\dfrac{3}{13}}$

➜ B's Sacrifice = $\sf{\dfrac{13 - 12}{52}}$

➜ B's Sacrifice = $\sf{\dfrac{1}{52}}$

Thus,

• ✪ Sacrificing Ratio between A and B = A's Sacrifice : B's Sacrifice

➨ Sacrificing Ratio between A and B = $\sf{\dfrac{3}{52}}$ : $\sf{\dfrac{1}{52}}$

∴ Sacrificing Ratio between A and B = 3 : 1.