1. Algebraic Derivation of Reverse Sales Tax
When tax is added to a transaction, the gross price ($G$) is equal to the net price ($N$) plus the tax amount ($T$).
This derivation proves why dividing by $(1 + r)$ is mathematically required when reversing sales tax, VAT, GST, or HST.
2. Complete Formula Reference Table
| Calculation Type | Known Inputs | Target Output | Exact Formula |
|---|---|---|---|
| Standard Reverse Tax (Sales Tax / VAT / GST) | Gross Total & Tax Rate % | Pre-Tax Net Subtotal | Net = Gross ÷ (1 + Tax Rate Decimal) |
| Reverse Tax Amount Paid | Gross Total & Tax Rate % | Tax Amount Paid ($) | Tax = Gross - [Gross ÷ (1 + Tax Rate)] |
| Tax Rate Solver | Gross Total & Net Subtotal | Tax Rate (%) | Rate % = [(Gross ÷ Net) - 1] × 100 |
| Payroll Net to Gross Salary | Target Net & Effective Tax Rate % | Required Gross Salary | Gross = Net ÷ (1 - Effective Rate Decimal) |
| Reverse Tax & Tip Split | Gross Total, Tax % & Tip % | Pre-Tax Net Food Subtotal | Net = Gross ÷ (1 + Tax Rate + Tip Rate) |
3. Solving for Tax Rate Given Net & Gross
If you know both the pre-tax net price ($N$) and the tax-included gross total ($G$), you can solve for the tax rate percentage ($r\%$) using:
Worked Example: If an invoice lists a net price of $100.00 and a total price of $115.00:
Rate = [($115.00 ÷ $100.00) - 1] × 100 = (1.15 - 1) × 100 = 15%.
4. Canadian Dual Tax Rules (GST + PST / QST)
In dual-tax jurisdictions like Quebec or British Columbia, two taxes apply to the transaction.
A. Additive Rule (Modern Quebec QST & BC PST)
Gross = Net × (1 + rate1 + rate2)
Net = Gross ÷ (1 + 0.05 GST + 0.09975 QST) = Gross ÷ 1.14975
B. Historical Compounded Rule
Gross = Net × (1 + rate1) × (1 + rate2)
5. Payroll Net-to-Gross Salary Formula
Unlike flat sales taxes where tax is added to the net ($G = N \times (1 + r)$), income taxes are withheld as a percentage of the gross salary ($N = G \times (1 - t)$).