Ratio: Concept, Properties & Business Distribution

1. Meaning & Commercial Definition of Ratio

A ratio is a relationship between two numbers or quantities of the same kind, expressing how many times one contains or is contained within the other. It is written as a : b (read as “a is to b”), or as the fraction a / b:

• The first quantity a is called the Antecedent.
• The second quantity b is called the Consequent (b ≠ 0).

Fundamental Operational Properties of Ratios:

1. Unitless Dimension: A ratio is a pure number; it carries no physical unit of measurement.
2. Homogeneity of Units: Quantities compared must be converted into the identical unit before computing the ratio (e.g., comparing 50 paise to ₹2 requires converting ₹2 to 200 paise, yielding 50 : 200 = 1 : 4).
3. Invariance under Multiplication/Division: Multiplying or dividing both antecedent and consequent by the same non-zero constant does not alter the value:

   (a × k) : (b × k) = a : b  (k ≠ 0)

2. Taxonomy of Special Mathematical Ratios

3. Commercial Applications: Division in Given Ratio

In corporate partnerships, profits, losses, or assets are distributed in proportion to capital invested and time duration:

Problem 1: Partnership Profit Sharing

Question: A, B, and C invest ₹50,000, ₹75,000, and ₹1,25,000 respectively in a commercial venture. At the end of the financial year, the net profit is ₹72,000. Distribute the profit among the three partners.

Step-by-Step Solution:

1. Determine the capital investment ratio:
   • A : B : C = 50,000 : 75,000 : 1,25,000
   • Divide through by 25,000:
   • A : B : C = 2 : 3 : 5
2. Sum of the ratio terms:
   • 2 + 3 + 5 = 10
3. Compute each partner’s share:
   • A’s Share: (2 ÷ 10) × ₹72,000 = ₹14,400
   • B’s Share: (3 ÷ 10) × ₹72,000 = ₹21,600
   • C’s Share: (5 ÷ 10) × ₹72,000 = ₹36,000
4. Verification:
   • 14,400 + 21,600 + 36,000 = ₹72,000 (Exact match with total profit!)