The Master Formulas
1. Total Price = $Base \times (1 + \frac{R}{100})$2. Base Price = $\frac{Total}{1 + \frac{R}{100}}$3. GST Amount = $Total - \frac{Total}{1 + \frac{R}{100}}$
Dimensional Analysis
Since the rate $R$ is a percentage (dimensionless), the term $(1 + R/100)$ is a pure number. Therefore, [Total Price] = [Base Price], which are both units of Currency ($M L^2 T^{-2}$ in a stretch, but usually just a monetary unit). This confirms the linear scaling property.
Variations: GST Split
For intra-state sales in India:$$CGST = SGST = \frac{1}{2} \times \text{Total GST}$$This is a simple Bifurcation Rule. The total tax is divided equally, but the base remains the same.
Shortcuts & Mnemonics
- The "100 + R" Rule: To find the base price, always divide the total by $(100 + R)$ and multiply by $100$.Example: For $12\%$ tax, Base = $(Total / 112) \times 100$.
- Mnemonic: "Inside out, Divide; Outside in, Multiply." (Divide to find the internal Base price; Multiply to find the external Total price).
Edge Cases
- $0\%$ GST (Exempt): The Multiplier is $1$. Total = Base. (Used for essential goods).
- Composite Supply: If two items with different tax rates are sold together, the highest rate usually applies to the whole bundle. This is a Min-Max Constraint problem.
- Negative Tax (Subsidy): If the rate $R$ is negative, the "Tax" becomes a subsidy, and the final price is lower than the base price ($Total = Base \times 0.90$ for a $10\%$ subsidy).
