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Proportion: Laws, Types & Chain Rule Applications | B.Com Notes

Study material and explanations for Unit 1

Proportion: Laws, Types & Chain Rule Applications

1. Meaning & Core Law of Proportion

An equality of two ratios is termed a Proportion. If two ratios a : b and c : d are equal, the four quantities are in proportion:

a : b = c : d  (or written as a : b :: c : d)

Where:

  • a and d are called the Extremes (outer terms)
  • b and c are called the Means (inner terms)

The Fundamental Law of Proportion:

Product of Extremes = Product of Means
a × d = b × c

From this rule, any unknown fourth term can be computed directly:

  • d = (b × c) ÷ a
  • a = (b × c) ÷ d

2. Classification of Proportions

The Three Major Forms of Proportion
1. Direct Proportion
An increase in one quantity produces a proportionate increase in the other (y / x = constant k).
Example: More units purchased → higher total invoice bill.
2. Inverse Proportion
An increase in one quantity produces a proportionate decrease in the other (x × y = constant k).
Example: More workers deployed → fewer days required to finish a job.
3. Continued Proportion
Three quantities a, b, c are in continued proportion if a : b = b : c.
b² = a × c  ⟹  b = √(a × c)
b is called the Mean Proportional between a and c.

3. Commercial Chain Rule Applications

The Chain Rule solves complex industrial multi-variable problems by analyzing whether each variable relates directly or inversely to the target unknown:

Problem:

Question: If 15 factory workers can assemble 180 electronic components in 8 days working 6 hours a day, how many days will 20 workers take to assemble 300 components working 9 hours a day?

Step-by-Step Solution:

  1. Compare each factor individually with the unknown (Days, D):

    • Workers (15 → 20): More workers require fewer days (Inverse proportion) → Factor = 15 / 20
    • Hours per day (6 → 9): Working more hours per day requires fewer days (Inverse proportion) → Factor = 6 / 9
    • Components (180 → 300): Assembling more components requires more days (Direct proportion) → Factor = 300 / 180

  2. Set up the compound proportion Chain Rule equation:

    • D = 8 × (15 / 20) × (6 / 9) × (300 / 180)

  3. Simplify each fraction individually:

    • 15 / 20 = 3 / 4
    • 6 / 9 = 2 / 3
    • 300 / 180 = 5 / 3

  4. Multiply the simplified fractions:

    • D = 8 × (3 / 4) × (2 / 3) × (5 / 3)
    • Numerator product: 8 × 3 × 2 × 5 = 240
    • Denominator product: 4 × 3 × 3 = 36
    • D = 240 ÷ 36 = 6.67 days

  5. Conclusion:

    • 20 workers will take 6.67 days (or 6 days and 16 hours) to assemble the 300 components.