Discounts reduce an original price by a specified amount or percentage.
A 20% discount does not mean you pay 20% of the original price. It means you save 20% and pay the remaining 80%.
That distinction is the key to calculating sale prices accurately.
For most percentage discounts, you can either calculate the amount saved and subtract it from the original price or multiply the original price directly by the percentage that remains.
Discount Formula
The discount amount is:
Discount Amount = Original Price × Discount Rate
The final price is:
Final Price = Original Price − Discount Amount
You can also calculate the final price directly:
Final Price = Original Price × (1 − Discount Rate)
Percentages should be converted to decimals before multiplication.
For example:
25% = 0.25
Discount Example
Suppose an item costs $120 and is discounted by 25%.
First calculate the discount amount:
Discount = $120 × 0.25
Discount = $30
Now subtract it:
Final Price = $120 − $30
Final Price = $90
The customer saves $30 and pays $90.
Direct Final-Price Method
Because a 25% discount leaves 75% of the price:
100% − 25% = 75%
Then:
Final Price = $120 × 0.75
Final Price = $90
Both approaches produce the same answer.
10% Discount Example
Suppose an item costs $85.
Discount:
$85 × 10% = $8.50
Final price:
$85 − $8.50 = $76.50
Therefore:
- Original price = $85
- Savings = $8.50
- Final price = $76.50
35% Discount Example
Suppose the original price is $240.
Discount = $240 × 0.35
Discount = $84
Then:
Final Price = $240 − $84
Final Price = $156
Alternatively:
Final Price = $240 × 0.65
Final Price = $156
How to Calculate Percent Off From Two Prices
Sometimes the original price and sale price are known, but the discount percentage is not.
Use:
Percent Discount = (Original Price − Sale Price) ÷ Original Price × 100
Suppose:
- Original price = $200
- Sale price = $150
Savings:
$200 − $150 = $50
Discount percentage:
$50 ÷ $200 × 100
Discount = 25%
The item is 25% off.
How to Find the Original Price From a Discounted Price
Suppose an item is 20% off and its sale price is $80.
A 20% discount means the customer pays 80% of the original price.
Sale Price = Original Price × 0.80
Rearrange:
Original Price = Sale Price ÷ 0.80
Original Price = $80 ÷ 0.80
Original Price = $100
The original price was $100.
Reverse Discount Formula
In general:
Original Price = Final Price ÷ (1 − Discount Rate)
If a $136 item reflects a 15% discount:
Original Price = $136 ÷ 0.85
Original Price = $160
This is different from simply adding 15% to $136.
Why?
Because the original 15% discount was calculated from the larger original price, not from the smaller sale price.
Why Adding the Discount Back Does Not Work
Suppose $100 is reduced by 20%.
$100 × 0.80 = $80
If you then increase $80 by 20%:
$80 × 1.20 = $96
You do not return to $100.
To reverse the original discount:
$80 ÷ 0.80 = $100
Percentage decreases and equal percentage increases are not symmetrical because they use different bases.
Multiple Discounts
Successive discounts should normally be applied one after another, not simply added.
Suppose an item costs $200 and receives:
- 20% off;
- then an additional 10% off.
First discount:
$200 × 0.80 = $160
Second discount:
$160 × 0.90 = $144
Final price:
$144
Total savings:
$200 − $144 = $56
Effective discount:
$56 ÷ $200 × 100 = 28%
The combined discount is 28%, not 30%.
Combined Discount Formula
For successive discounts:
Final Price = Original Price × (1 − d₁) × (1 − d₂)
For 20% and then 10%:
Final Price Factor = 0.80 × 0.90
Final Price Factor = 0.72
Therefore, the customer pays 72% of the original price and saves:
100% − 72% = 28%
Three Successive Discounts
Suppose an item gets:
- 20% off;
- another 15% off;
- another 5% off.
Remaining price factor:
0.80 × 0.85 × 0.95 = 0.646
The customer pays 64.6% of the original price.
Effective discount:
100% − 64.6% = 35.4%
The three discounts combine to 35.4% off, not 40%.
Fixed-Dollar Discounts
Not every discount is expressed as a percentage.
Suppose a coupon gives $25 off a $140 purchase.
Final Price = $140 − $25
Final Price = $115
To express the $25 savings as a percentage:
Discount Percentage = $25 ÷ $140 × 100
Discount Percentage ≈ 17.86%
Minimum-Purchase Discounts
Suppose an offer gives $20 off purchases of at least $100.
On a $100 purchase:
Effective Discount = $20 ÷ $100 = 20%
On a $200 purchase:
Effective Discount = $20 ÷ $200 = 10%
The same fixed-dollar coupon therefore produces different effective percentages depending on purchase size.
Discount Before Sales Tax
Where tax rules require sales tax to be applied to the discounted selling price, the sequence can be modeled as:
Discounted Price = Original Price × (1 − Discount Rate)
Then:
Final Total = Discounted Price × (1 + Tax Rate)
Suppose:
- Original price = $100
- Discount = 20%
- Illustrative tax rate = 8%
Discounted price:
$100 × 0.80 = $80
Tax:
$80 × 0.08 = $6.40
Total:
$80 + $6.40 = $86.40
Actual tax treatment depends on the applicable jurisdiction and transaction.
Discounts and Depreciation
A purchase discount—in the general sense of paying less than a listed amount—and accounting depreciation should not be confused.
For an asset, the acquisition price or recognized cost is established first according to applicable accounting treatment.
Depreciation then allocates the depreciable amount across the asset’s useful life.
A 20% retail discount is therefore not the same thing as a 20% annual depreciation rate.
Discounts and Current Yield
Discount terminology also appears in fixed-income markets.
A bond can trade below face value, but current yield still uses:
Current Yield = Annual Coupon ÷ Market Price
A bond priced at a discount is not analyzed simply by calculating “percent off face value” and treating that percentage as yield.
The return relationship also depends on coupon income and potentially maturity value.
Discounts and Dividend Yield
Dividend yield is another percentage based on market price.
It measures annual dividends relative to share price.
A stock falling in price can mechanically increase dividend yield if the dividend remains unchanged, but that is not equivalent to a retail discount guaranteeing greater value.
Market prices can fall because underlying expectations have changed.
Discounts and Currency Exchange
If a purchase occurs in another currency, calculate the currency exchange separately.
For example, an item might first receive a 20% discount in euros and then be converted into dollars.
The discount determines the local-currency sale price.
The exchange rate determines its equivalent value in the other currency.
Combining the two steps incorrectly can distort the final amount.
Discounts and Dollar-Cost Averaging
Dollar-cost averaging involves investing fixed amounts over time rather than calculating consumer sale discounts.
When market prices fall, a fixed investment amount can purchase more units, but calling every market decline a “discount” introduces a valuation judgment that the percentage-off formula cannot establish.
A lower market price is not automatically evidence that an investment is undervalued.
Percentage Points vs Percent Discounts
A change from 30% off to 40% off is an increase of:
40% − 30% = 10 percentage points
But the discount rate itself increased relative to the original 30% by:
(40% − 30%) ÷ 30% × 100
33.33%
Percentage points and percentage changes are different measurements.
Comparing Two Sale Offers
Suppose Store A offers 25% off a $120 item.
Final Price = $120 × 0.75
Final Price = $90
Store B lists the same item at $100 and offers 10% off.
Final Price = $100 × 0.90
Final Price = $90
Both offers produce the same $90 final price.
This is why comparing final prices is often more useful than comparing headline discount percentages.
Common Discount Mistakes
One mistake is paying the discount percentage instead of the remaining percentage.
For 30% off, you pay 70%:
Final Price = Original Price × 0.70
Another mistake is adding successive discounts together.
A third is trying to reverse a discount by adding the same percentage to the reduced price.
Finally, comparing discount percentages without comparing original prices can make one offer appear better when the final prices are identical.
Frequently Asked Questions
What is a discount?
A discount is a reduction from an original or listed price.
What is the discount formula?
Discount Amount = Original Price × Discount Rate
How do I calculate the final price?
Final Price = Original Price × (1 − Discount Rate)
What is 20% off $100?
$100 × 0.80 = $80
The final price is $80 and the savings are $20.
What is 30% off $250?
$250 × 0.70 = $175
The final price is $175.
How do I calculate the discount percentage from two prices?
Discount % = (Original Price − Sale Price) ÷ Original Price × 100
How do I find the original price from the sale price?
Original Price = Sale Price ÷ (1 − Discount Rate)
Do two 20% discounts equal 40% off?
No.
0.80 × 0.80 = 0.64
You pay 64% of the original price, so the effective discount is 36%.
Is $20 off the same as 20% off?
Only when the original price is $100. A fixed-dollar discount produces a different percentage depending on the original price.
Why can’t I add 20% to reverse a 20% discount?
Because the decrease and increase use different starting values. Divide by 0.80 to reverse a 20% reduction.
Should I compare discount percentages or final prices?
Final price is usually the more direct measure when comparing the actual cost of two offers.
Why do discounts matter in financial planning?
Understanding percentage reductions helps with budgeting and purchase comparisons within the broader Savings & Investing framework.
