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Numerical Questions · Q2

Q.A, B, C were partners in a firm sharing profits in 3:2:1 ratio. They admitted D for 10% profits. Calculate the new profit sharing ratio?

Sikkim CbseNCERTSubjective· 3mImportance★★★★★
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✓ Free question

The new profit-sharing ratio among A, B, C, and D is 9:6:3:2, calculated by first determining the remaining share for old partners after D’s admission and then redistributing it in their old ratio.

Concept: Admission of a Partner – Adjusting the Profit-Sharing Ratio

When a new partner is admitted, the old partners must sacrifice a portion of their profit share to accommodate the newcomer. The key rule is: the new partner’s share is taken from the old partners in their existing profit-sharing ratio, unless otherwise agreed. Here, D is admitted for 10% (i.e., 1/10th) of the profits. The old partners A, B, and C shared profits in the ratio 3:2:1.

The logic is straightforward: the total profit is always 1 (or 100%). After D takes his 10%, the remaining 90% (or 9/10) is to be shared among A, B, and C in their old ratio of 3:2:1. We then express each old partner’s new share as a fraction of the total, and finally convert all fractions into a common denominator to get the new ratio.

Watch out

Common Mistake

Do not simply add D’s share to the old ratio (e.g., 3:2:1:1). That would be wrong because D’s 10% is not a part of the old 3:2:1 ratio; it is a separate slice taken from the whole. Always first reduce the old partners’ total share to (1 – new partner’s share), then redistribute.

Solution: Step-by-Step Calculation

Step 1: Determine the total share of old partners after D’s admission

Total profit = 1 (or 100%)

D’s share = 10% = 1/10

Remaining share for A, B, C = 1 – 1/10 = 9/10

Step 2: Distribute the remaining share among A, B, C in their old ratio (3:2:1)

Old ratio sum = 3 + 2 + 1 = 6

A’s new share = (3/6) of (9/10) = (3/6) × (9/10) = 27/60

B’s new share = (2/6) of (9/10) = (2/6) × (9/10) = 18/60

C’s new share = (1/6) of (9/10) = (1/6) × (9/10) = 9/60

D’s share = 1/10 = 6/60 (to make denominator common)

Step 3: Express all shares with a common denominator (60)

A = 27/60, B = 18/60, C = 9/60, D = 6/60

Step 4: Write the new profit-sharing ratio

New ratio = 27 : 18 : 9 : 6

Simplify by dividing each term by their common factor (3):

27 ÷ 3 = 9, 18 ÷ 3 = 6, 9 ÷ 3 = 3, 6 ÷ 3 = 2

Thus, the new ratio = 9 : 6 : 3 : 2

Tip

Shortcut

Instead of working with fractions, you can directly compute: New share of an old partner = (Old ratio numerator / Sum of old ratio) × (1 – New partner’s share). Then convert all to a common denominator. Here, (3/6)×(9/10) = 27/60, etc. The simplification to 9:6:3:2 is immediate.

Working Notes

PartnerOld Share (fraction)Calculation of New ShareNew Share (fraction)New Share (simplified)
A3/6(3/6) × (9/10) = 27/6027/609/20
B2/6(2/6) × (9/10) = 18/6018/606/20
C1/6(1/6) × (9/10) = 9/609/603/20
D–Given: 1/10 = 6/606/602/20

Check: Sum of new shares = 27/60 + 18/60 + 9/60 + 6/60 = 60/60 = 1 (or 100%). Correct.

✓Final answer

The new profit-sharing ratio among A, B, C, and D is 9:6:3:2. This matches the official NCERT answer key.

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