Progressive taxation assesses income in ascending rate tiers, ensuring only earnings within each bracket are taxed at that tier percentage.

### Mathematical Formula Federal income tax sums liabilities across each applicable marginal bracket: $$\text{Federal Tax} = \sum_{i=1}^{k} (\min(\text{Income}, \text{Max}_i) - \text{Min}_i) \times \text{Rate}_i$$ $$\text{State Tax} = \text{Income} \times \text{State Rate}$$ $$\text{Effective Tax Rate} = \left( \frac{\text{Federal Tax} + \text{State Tax}}{\text{Income}} \right) \times 100$$ $$\text{Net Take-Home Pay} = \text{Income} - \text{Federal Tax} - \text{State Tax}$$

### Worked Numeric Example A single filer earning **$75,000 annual gross salary** in California (where the calculator applies a 9.3% benchmark state rate) evaluates their tax liability as follows: 1. Tier 1 ($0 to $11,600 at 10%): $1,160.00. 2. Tier 2 ($11,600 to $47,150 at 12%): $4,266.00 ($35,550 × 0.12). 3. Tier 3 ($47,150 to $75,000 at 22%): $6,127.00 ($27,850 × 0.22). 4. Total Federal Income Tax: **$11,553.00** (Marginal Rate: 22%). 5. California State Tax (9.3%): $75,000 * 0.093 = **$6,975.00**. 6. Combined Tax Liability: $11,553.00 + $6,975.00 = **$18,528.00** (a 24.70% effective combined tax rate), leaving **$56,472.00 in annual net take-home pay ($4,706.00 per month)**.