Data Source and Methodology
Authoritative Data Source: NIST/SEMATECH e-Handbook of Statistical Methods — Section 1.3.5 Percentages (2012). National Institute of Standards and Technology (NIST). Direct link to source.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Single percentage discount:
Discount = P \times \frac{d}{100}
Final = P - Discount
Flat discount:
Final = P - A
Stacked percentage discounts (e.g., d₁, d₂, d₃):
P_{\text{after %}} = P \times \prod_{i}(1 - \frac{d_i}{100})
Final = \max\!\big(0,\; P_{\text{after %}} - A_{\text{flat}}\big)
Sales tax applied after discounts:
Final_{\text{with tax}} = Final \times \left(1 + \frac{t}{100}\right)
Effective discount (including flats):
d_{\text{eff}} = 1 - \frac{Final}{P}
Reverse (recover original P from Final_with_tax):
P = \frac{ \frac{Final_{\text{with tax}}}{1 + t/100} + A_{\text{flat}} }{ \prod_{i}(1 - d_i/100) }
Glossary of Variables
- P: Original price per unit.
- d, d₁, d₂, d₃: Discount percentages applied sequentially.
- A_flat: Total flat discount subtracted after percentage discounts (includes “Flat off” and Discount 1 if set to amount).
- t: Sales tax percentage applied after all discounts.
- Final: Price per unit after all discounts, before tax.
- Final_with_tax: Price per unit after discounts and tax.
- d_eff: Effective overall discount percentage relative to P.
- Quantity: Number of units; totals scale linearly with quantity.
How It Works: A Step-by-Step Example
Suppose P = $120, Discount 1 = 25% (percent), Discount 2 = 10%, Additional flat off = $5, and Sales tax t = 8.5%.
- Percent stage: P_after% = 120 × (1 − 0.25) × (1 − 0.10) = 120 × 0.75 × 0.90 = 81.00.
- Flat stage: Final = max(0, 81.00 − 5) = 76.00.
- Tax stage: Final_with_tax = 76.00 × (1 + 0.085) = 82.46.
- Effective discount: d_eff = 1 − 76/120 = 36.67%.
If you only know the final with tax ($82.46) and want the original price, reverse the steps using the formula above.
Frequently Asked Questions (FAQ)
Is the effective discount just the sum of individual percentages?
No. Stacked discounts multiply their complements: 1 − (1 − d₁)(1 − d₂)… This yields a smaller value than simple addition unless one of the discounts is zero.
How are flat discounts combined with percentage discounts?
Percentages are applied first to P. Then the total flat-off amount is subtracted from the percentage-discounted price. Tax, if any, is applied last.
Can the price become negative after discounts?
No. The tool caps the discounted price at zero and highlights inputs if the flat discount is too large.
Does the calculator support reverse-calculation with tax?
Yes. The calculator removes tax first (if provided) and then reverses stacked discounts and flat offs to estimate the original price.
What level of numeric precision is used?
Values are computed in full floating-point precision and displayed rounded to two decimal places for currency amounts.
Who is this tool for?
Consumers validating sale prices, store associates applying promos, and analysts modeling blended promotions with tax.