1.1 The Concept of Ratio (অনুপাতের ধারণা)
A ratio is a comparative division of two quantities of the same kind measured in the same physical unit. The ratio of quantity $a$ to quantity $b$ ($b eq 0$) is written as $a : b = rac{a}{b}$.
- Antecedent (পূর্বপদ): The first term $a$.
- Consequent (উত্তরপদ): The second term $b$.
- Dimensionless Property: A ratio is an abstract pure real number and carries no physical units.
1.2 Classification of Ratios
| Ratio Type | Condition | Fraction Value | Example |
|---|---|---|---|
| Ratio of Equality (সমানুপাত / সাম্যানুপাত) | $a = b$ | $ rac{a}{b} = 1$ | $5 : 5 = 1 : 1$ |
| Ratio of Greater Inequality (গুরু অনুপাত) | $a > b$ | $ rac{a}{b} > 1$ | $7 : 4$ (Antecedent exceeds consequent) |
| Ratio of Lesser Inequality (লঘু অনুপাত) | $a < b$ | $ rac{a}{b} < 1$ | $3 : 8$ (Antecedent is less than consequent) |
| Inverse Ratio (ব্যস্ত বা বিপরীত অনুপাত) | Interchanging terms | $b : a = rac{b}{a}$ | Inverse of $4 : 9$ is $9 : 4$ |
1.3 Derived Ratios & Continued Ratios
- Compound / Mixed Ratio (মৌলিক বা মিশ্র অনুপাত): Formed by taking the product of antecedents to the product of consequents. For $a:b$ and $c:d$, the compound ratio is $(a \cdot c) : (b \cdot d)$.
- Duplicate Ratio (দ্বিগুণানুপাত): $a^2 : b^2$.
- Sub-duplicate Ratio (দ্বিভাজিত অনুপাত): $\sqrt{a} : \sqrt{b}$.
- Triplicate Ratio (ত্রিগুণানুপাত): $a^3 : b^3$.
- Sub-triplicate Ratio (ত্রিভাজিত অনুপাত): $\sqrt[3]{a} : \sqrt[3]{b}$.
- Continued Ratio (ধারাবাহিক অনুপাত $a:b:c$): If $a:b = 2:3$ and $b:c = 4:5$, equate the common term $b$ by taking $ ext{LCM}(3, 4) = 12$: $a:b = 8:12$ and $b:c = 12:15 \implies a:b:c = 8:12:15$.