Worked Example: Discount and Tax Bases on a $100 Item

Apply two percentage discounts

A $100 item receives 20% off, leaving $80, then 10% off that amount, leaving $72. Savings before tax are $28, or 28% of the original price. Adding the two discount rates would incorrectly predict $70.

Apply 8% tax to the discounted base

Using a discounted-price base, tax is $72 × 0.08 = $5.76. The total is $77.76. A second calculation using total factor 0.8 × 0.9 × 1.08 reproduces the result.

Compare a different tax-base assumption

If the specified rule taxes the original $100, tax is $8 and the payable total is $80. The $2.24 difference is 8% of the $28 discount. This is a change in model assumptions, not a rounding error.

Add a fixed coupon carefully

A $10 coupon after the two percentage discounts leaves $62 before tax; applying it before both leaves $90 × 0.8 × 0.9 = $64.80. Percentage factors commute with each other, but a subtraction does not generally commute with them. Check the retailer’s order and any excluded amount.

Reconcile a receipt line by line

Identify the list price, current price, discount order, coupon treatment, taxable base, tax rate and line-level cent rounding. Shipping or fees may add another base. Keep a changed rule as a separate scenario; do not edit the original price merely to force agreement with the receipt.

A second checkout with three sequential percentage reductions

Use an independent illustrative item priced at $250. Apply 15% off, then another 10%, then another 5%, followed by 7.5% tax on the final discounted price. The remaining-price factors are 0.85, 0.90 and 0.95, whose product is 0.72675. The tax-free selling price is $181.6875, savings before tax are $68.3125, and the effective discount is 27.325%. Adding fifteen, ten and five would incorrectly claim a 30% reduction.

The discounted-price tax is $181.6875 × 0.075 = $13.6265625. Adding it gives $195.3140625, or $195.31 when the final amount is rounded once to cents. In the current discount form, enter original price 250, first discount 15, additional successive discounts 10, 5 and tax 7.5. The form retains intermediate arithmetic precision rather than imposing a retailer's line-by-line cent policy.

If a separate checkout rounds the selling price first to $181.69 and rounds tax on that price to $13.63, its total is $195.32. The one-cent difference is a specified rounding-order effect. It does not mean the percentage factors changed. A receipt audit needs both the discount order and the stages at which cents become final; two correctly implemented policies can differ when one preserves intermediate fractions and the other finalizes each line.

A third item makes the tax-base difference visible

Take a separate $129.99 item with 25% off followed by 20% off. The paid fraction is 0.75 × 0.80 = 0.60, so the tax-free amount is $77.994 and savings are $51.996. The effective discount is 40%, not 45%. With 8.25% tax on the discounted amount, the exact tax is $6.434505 and the total is $84.428505, reported as $84.43 after one final cent rounding.

For a deliberately different illustrative rule, apply the same 8.25% rate to the original $129.99. Tax is then $10.724175 and the payable total is $88.718175, or $88.72 under the same final-rounding convention. The exact difference between the two totals is $4.28967, which also equals the discount savings $51.996 multiplied by 0.0825. That second expression independently verifies the change in taxable base.

The current tool's tax field expressly means tax on the discounted price. It reproduces the first scenario directly. The original-price tax scenario is a manual comparison, not a hidden alternate setting in the form. Neither example states a jurisdiction's law. Discounts, manufacturer coupons, retailer coupons, shipping and fees can receive different treatment under actual rules, so the arithmetic inputs must be chosen from the applicable policy rather than inferred from these illustrative rates.

Distinguish fixed coupons, effective rates and receipt-level rounding

On the third item, a hypothetical $10 fixed coupon applied after both percentage reductions leaves $67.994 before tax. Applying the coupon first gives ($129.99 − $10) × 0.60 = $71.994. The four-dollar difference occurs because the coupon itself is reduced by the later percentage factors in the second order. Percentage discounts commute when they affect the same eligible base, but fixed subtraction and multiplication do not generally commute.

Fixed coupons are not a field in the current discount form. Reproduce a before-discount coupon scenario by adjusting the price only if that is the intended eligible base, and label the adjustment explicitly. For an after-discount coupon, calculate the percentage-only amount first and subtract the coupon separately. Keep those manual extensions separate from what the form directly validates. Offer minimums, exclusions or caps introduce additional rules that the simple multiplication model does not infer.

To compare offers, save the original eligible price, every percentage rate, the order of fixed adjustments, the tax base, tax rate and rounding stages. Effective discount describes savings relative to the original tax-free price; it is not the sum of the displayed promotional percentages and need not equal savings measured against a tax-inclusive reference. Choose one denominator and label it.

These three cases demonstrate reproducible checkout arithmetic rather than claiming observed retailer prices or customer transactions. The calculation tests validate the factors and the stated tax bases. An actual tax classification, coupon eligibility or receipt policy requires independent verification. A difference larger than a cent often points to a changed base or an omitted charge; a one-cent difference may reflect a finalization rule, but that explanation should be demonstrated from the ledger rather than guessed.