Backing Out Sales Tax From a Receipt: Why T * (1 - r) Is Always Wrong

When an accounting script or expense parser tries to recover the pre-tax price from a tax-inclusive receipt total, the most common bug is multiplying the total by (1 - r) instead of dividing by (1 + r).

Forward vs. Reverse Arithmetic

If the pre-tax price is P = 100.00 and the combined tax rate is r = 0.0825 (8.25%):

Total = P * (1 + r) = 100.00 * 1.0825 = 108.25 

Going backward from Total = 108.25:

Wrong: 108.25 * (1 - 0.0825) = 99.319375 -> $99.32 (off by $0.68) Right: 108.25 / (1 + 0.0825) = 100.00 

Because (1 + r)(1 - r) = 1 - r^2, subtracting r introduces a systematic downward bias of r^2 * P. At an 8.25% tax rate that is 0.68% of every invoice; at 10% it is 1% of gross spend.

Two Edge Cases in U.S. Sales Tax Code

1. Integer-cent rounding per line vs. per invoice. Point-of-sale terminals round round_half_up(subtotal * r) to integer cents. When an invoice contains multiple taxable lines, summing rounded line-level taxes can differ by $0.01 from applying r to the combined subtotal.
2. Jurisdiction stacking. State, county, city, and transit-district rates combine into one effective rate r, though certain states cap local tax above a threshold per single item.

If you want to inspect forward tax and reverse pre-tax extraction across U.S. state rates in integer cents, the Miryori Sales Tax Calculator shows the exact breakdown step by step.

Related reading

• a shorter math note on why reverse sales tax diverges quadratically

YMYL note: examples above are illustrative arithmetic only; verify jurisdiction-specific exemptions, thresholds, and filing rules with local tax authorities.