Properties of ratio and proportion allow systematic manipulation of ratios and proportions to solve real-world business problems involving cost allocation, profit sharing, and resource distribution.
## Core concept
A ratio is a comparison of two quantities using division. A proportion is an equality of two ratios.
Properties of Ratio: - If a : b = c : d, then the ratios are equal - Ratios remain unchanged if both terms are multiplied or divided by the same non-zero number - a : b = ka : kb (scaling property) - a : b = a/k : b/k (reduction property)
Properties of Proportion: If a : b = c : d (written as a/b = c/d), then: - Invertendo: b : a = d : c - Alternando: a : c = b : d - Componendo: (a + b) : b = (c + d) : d - Dividendo: (a − b) : b = (c − d) : d - Componendo & Dividendo: (a + b) : (a − b) = (c + d) : (c − d)
## Formulas and rules
| Property | Mathematical form | Application | |----------|-------------------|-------------| | Invertendo | If a/b = c/d, then b/a = d/c | Reversing denominators and numerators | | Alternando | If a/b = c/d, then a/c = b/d | Rearranging terms within proportions | | Componendo | If a/b = c/d, then (a+b)/b = (c+d)/d | Adding numerators to denominators | | Dividendo | If a/b = c/d, then (a−b)/b = (c−d)/d | Subtracting numerators from denominators | | Componendo & Dividendo | If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d) | Combined operation for elegant simplification |
## Common exam applications
Profit Sharing: Two partners A and B contributed capital in ratio 3 : 5. If total profit is ₹4,000, find each share using componendo.
- A : B = 3 : 5
- Using componendo: (A + B) : B = (3 + 5) : 5 = 8 : 5
- Total = 8 parts = ₹4,000
- One part = ₹500
- A's share = 3 × 500 = ₹1,500; B's share = 5 × 500 = ₹2,500
Cost Allocation: If material and labour costs are in ratio 7 : 3, and their difference is ₹2,000, find each cost using dividendo. - M : L = 7 : 3 - Using dividendo: (M − L) : L = (7 − 3) : 3 = 4 : 3 - M − L = ₹2,000 = 4 parts - One part = ₹500 - Material = 7 × 500 = ₹3,500; Labour = 3 × 500 = ₹1,500
## Common mistakes
- Confusing invertendo with alternando: Invertendo reverses numerator and denominator; alternando swaps positions across the equal sign.
- Applying componendo without checking the original ratio: Properties work only with valid proportions; verify a/b = c/d before using.
- Sign errors in dividendo: When subtracting, ensure (a − b) > 0; if not, the logic may need adjustment.
- Not recognizing when to use componendo & dividendo: This combined property is powerful when dealing with sums and differences simultaneously; avoid solving two equations separately when one elegant operation suffices.
- Misapplying scaling: Multiplying only one term by k destroys the ratio; both terms must be scaled identically.
Exam tip: Proportion properties reduce calculation steps significantly. In divisional allocation and partnership problems, componendo & dividendo often yields answers in two lines instead of three-step algebra.