Reverse Percentage Formula: How to Find Original Price
One of the most persistent mathematical errors in retail shopping, invoicing, and small-business accounting is attempting to reverse a discount by simply adding the percentage back to the sale price. If an item is on sale for $80 after a 20% discount, adding 20% back gives $96, not the original $100. Because percentage increases and decreases scale to different base numbers, recovering pre-sale prices or backing out sales tax requires reverse percentage division.
To find the original price before a discount, divide the sale price by (1 − Discount Rate as a decimal): Original Price = Sale Price / (1 − d). For example, if an item costs $70 after a 30% discount, the original price was $70 / (1 − 0.30) = $70 / 0.70 = $100.00.
The "Add-Back" Fallacy: Why Adding the Percentage Back Fails
Why does adding 20% back to an $80 discounted price fail to return $100? The breakdown lies in the base value that the percentage scales against:
- The Forward Discount (Base = $100): A 20% discount on $100 deducts $20 ($100 × 0.20 = $20). The resulting sale price is $80.00.
- The Erroneous Reversal (Base = $80): If you calculate 20% of the $80 sale price, you are evaluating 20% of a smaller base number ($80 × 0.20 = $16). Adding $16 to $80 gives $96.00, leaving you $4 short of the true original price.
Because the base shifts from $100 down to $80, a 20% reduction requires a 25% increase ($16 / $80 = 0.20, but $20 / $80 = 0.25) to restore the starting figure.
The Step-by-Step Reverse Percentage Formula
To find the pre-discount price algebraically, express the transaction as a forward multiplier equation and solve for the unknown original price ($P_{\text{orig}}$):
Sale Price = Original Price × (1 − Discount Rate) // Dividing both sides by (1 − Discount Rate) yields: Original Price = Sale Price / (1 − Discount Rate)
Worked Real-World Examples:
Example 1: Finding the Original Price After a 20% Discount
A winter jacket is marked on sale for $64.00 after a 20% discount. What was the original price?
- Convert the percentage to a decimal: 20% = 0.20.
- Subtract the discount from 1: $1 - 0.20 = \mathbf{0.80}$.
- Divide the sale price by 0.80:
Original Price = $64.00 / 0.80 = $80.00
Example 2: Reversing a 35% Clearance Markdown
A pair of running shoes is purchased on clearance for $52.00 at 35% off.
- Convert 35% to a decimal: 0.35.
- Calculate the multiplier: $1 - 0.35 = \mathbf{0.65}$.
- Divide the sale price:
Original Price = $52.00 / 0.65 = $80.00
Reverse Percentage Multiplier & Reciprocal Matrix
Use this reference matrix to reverse standard retail discounts quickly. You can either divide by the decimal divisor or multiply by the reciprocal factor:
| Discount Rate (%) | Formula Divisor (1 − d) | Reciprocal Multiplier | Example ($100 Original → Sale) |
|---|---|---|---|
| 10% Off | Divide by 0.90 | × 1.111 | $90.00 / 0.90 = $100.00 |
| 15% Off | Divide by 0.85 | × 1.176 | $85.00 / 0.85 = $100.00 |
| 20% Off | Divide by 0.80 | × 1.25 | $80.00 / 0.80 = $100.00 |
| 25% Off | Divide by 0.75 | × 1.333 | $75.00 / 0.75 = $100.00 |
| 30% Off | Divide by 0.70 | × 1.428 | $70.00 / 0.70 = $100.00 |
| 40% Off | Divide by 0.60 | × 1.666 | $60.00 / 0.60 = $100.00 |
| 50% Off (Half Price) | Divide by 0.50 | × 2.00 (Double) | $50.00 × 2 = $100.00 |
| 75% Off | Divide by 0.25 | × 4.00 (Quadruple) | $25.00 × 4 = $100.00 |
Reverse Sales Tax Formula: Backing Out Tax from Gross Receipts
In accounting, finance, and invoicing, business owners frequently need to determine the net price of an item when given a gross total that already includes sales tax or Value Added Tax (VAT).
The common mistake is multiplying the gross total by the tax rate ($T \times t$). Because tax was added to the net price, multiplying the gross total calculates tax on top of tax.
The Correct Reverse Tax Formula:
Tax Amount = Gross Total − Net Price
Example: Backing Out 20% VAT from a $120.00 Invoice:
- The Incorrect Calculation: $120.00 × 0.20 = $24.00 tax → Net = $96.00 (Overreports tax by $4)
- The Correct Reverse Formula:
1. Net Price = $120.00 / (1 + 0.20) = $120.00 / 1.20 = $100.00
2. Tax Amount = $120.00 − $100.00 = $20.00 exact tax
Explore Urban Mixo's Commercial Math Suite:
- Full Discount Calculator Hub — Custom discount percentages, stacked coupons, and tax calculations.
- Percentage Calculator Tool — Percentage change, relative variance, and ratio math.
- 15% Off Calculator — Pre-calculated retail lookup matrix and restaurant tip split calculator.
- 20% Off Calculator — Instant price lookup table ($5 to $500) and mental math shortcuts.
- 30% Off Calculator — Stacked coupon compounding and clearance savings.
- 50% Off Calculator — Half-price clearance and liquidation lookup table ($5 to $1,000).
- Margin vs. Markup Guide — Commercial retail profitability equations explained.
Frequently Asked Questions
Why can't you just add the discount percentage back to the sale price?
Adding the discount percentage back fails because the percentage is applied to a smaller base number. For example, 20% of $100 is $20, but 20% of the discounted $80 is only $16. Adding $16 returns $96 instead of the true $100 original price.
What is the formula to calculate original price before discount?
The formula is: Original Price = Sale Price / (1 − Discount Rate as a decimal). If an item costs $60 after a 25% discount, the original price was $60 / (1 − 0.25) = $60 / 0.75 = $80.00.
How do you back out sales tax from a receipt total?
To find the pre-tax price, divide the total receipt price by (1 + Tax Rate as a decimal). For example, a $108.00 total with 8% sales tax has a net pre-tax price of $108.00 / 1.08 = $100.00, meaning the tax was exactly $8.00.
What is the reverse multiplier for a 20% discount?
The decimal divisor for a 20% discount is 0.80. Alternatively, multiplying the sale price by the reciprocal factor of 1.25 gives the exact original price in a single multiplication step ($80 × 1.25 = $100).
How do you reverse a 50% discount?
To reverse a 50% discount, simply multiply the sale price by 2 (or divide by 0.50). If an item costs $45.00 on clearance at 50% off, the original price was $45.00 × 2 = $90.00.