Discounts: Definitions, Formulas, and Examples

What is a discount?

A discount is a specific type of percent decrease applied to the price of an item or service. Sellers use discounts to reduce the cost for buyers during sales or promotions.

When an item is discounted, the buyer pays less than the original advertised cost. The amount saved is the discount itself, which is typically given as a percentage of the original price.

Identify list price, discount and sale price

Calculating discounts requires understanding three distinct amounts and how they relate to one another.

The list price is the starting price, the discount is the amount removed, and the sale price is the final amount paid.

• List price: The original price of the item before any changes. This is also called the marked price or regular price.
• Discount: The total amount of money subtracted from the list price.
• Sale price: The final price the buyer pays after the discount is applied.
  

The fundamental relationship between these values is a simple subtraction equation:

$\text{Sale Price} = \text{List Price} - \text{Discount}$

Find a discount amount

Discounts are usually advertised as a percentage rate rather than a flat currency amount. To find the currency reduction, calculate the percent of a number using the list price.

$\text{Discount} = \text{List Price} \times \text{Discount Rate}$

The discount rate must be written as a fraction or a decimal before multiplying. For example, a $20\%$ discount rate is written as $0.20$ or $\dfrac{20}{100}$.

Once the discount amount is found, subtract it from the list price to find the sale price.

Use a sale-price multiplier

Finding the discount and then subtracting requires two steps. To calculate the sale price in one step, use a sale-price multiplier.

Because the list price represents $100\%$, subtracting the discount percentage gives the percentage the buyer actually pays.

$\text{Paying Percentage} = 100\% - \text{Discount Rate}$

Convert this paying percentage to a decimal to form the multiplier.

$\text{Sale Price} = \text{List Price} \times (1 - \text{Discount Rate})$

If an item is discounted by $25\%$, the buyer pays $75\%$ of the original price. The sale-price multiplier is $0.75$.

Find a discount rate

Sometimes the list price and sale price are known, and the unknown value is the percentage discount rate.

First, subtract the sale price from the list price to find the absolute discount amount.

$\text{Discount} = \text{List Price} - \text{Sale Price}$

Next, divide the discount amount by the list price to find the decimal rate, and multiply by $100$ to convert it to a percentage.

$\text{Discount Rate} = \left( \dfrac{\text{Discount}}{\text{List Price}} \right) \times 100\%$

Always divide by the original list price, never the new sale price.

Worked examples

These steps are foundational for solving all percentage word problems. Finding the original list price before a discount is an application of reverse percentages.

Example 1: Finding the sale price

Question: A jacket has a list price of $80$ dollars. It is on sale with a $15\%$ discount. What is the sale price?

Method:

1. Convert the discount percentage to a decimal.
2. Multiply the list price by the decimal to find the discount amount.
3. Subtract the discount amount from the list price.

Answer: The discount is $12$ dollars. The sale price is $68$ dollars.

Check: Use the multiplier method. $1 - 0.15 = 0.85$. Multiply $80 \times 0.85 = 68$ dollars. The answers match.

Example 2: Calculating the discount rate

Question: A bicycle originally costs $250$ dollars but is sold for $200$ dollars. What is the discount rate?

Method:

1. Subtract the sale price from the list price to find the discount amount.
2. Divide the discount amount by the list price.
3. Multiply by $100$ to convert to a percentage.

Answer: The discount is $50$ dollars. The rate is $\dfrac{50}{250} = 0.20$, which is $20\%$.

Check: Calculate a $20\%$ discount on $250$ dollars. $250 \times 0.20 = 50$ dollars. Subtracting $50$ dollars gives $200$ dollars.

Example 3: Finding the original list price

Question: After a $20\%$ discount, a smartphone is sold for $640$ dollars. What was the list price?

Method:

1. Determine the paying percentage. The buyer pays $100\% - 20\% = 80\%$ of the list price.
2. Write the relationship as an equation. $0.80 \times \text{List Price} = 640$.
3. Divide the sale price by the decimal multiplier to find the list price.

Answer: The list price was $\dfrac{640}{0.80} = 800$ dollars.

Check: Find $20\%$ of $800$ dollars, which is $160$ dollars. Subtract $160$ dollars from $800$ dollars to get $640$ dollars.

Common mistakes

Avoid these frequent errors when calculating discounts and sale prices.

• Using the sale price as the base: When calculating a percentage discount or finding the discount rate, the percentage must always refer to the original list price. A $10\%$ discount means $10\%$ of the list price, not $10\%$ of the final price.
• Subtracting percentages directly from prices: If an item costs $40$ dollars and has a $20\%$ discount, the sale price is not $40 - 20 = 20$ dollars. The $20\%$ must first be converted into a currency amount by multiplying $0.20 \times 40 = 8$ dollars, which is then subtracted.
• Adding successive discounts: If an item has a $20\%$ discount and then an additional $10\%$ discount, the total discount is not $30\%$. The second discount applies only to the new, already-reduced sale price.

Frequently asked questions

Discounts frequently appear in real-world math word problems. Review these common questions to solidify your understanding.

How do you calculate a discount?

Multiply the original list price by the percentage discount rate written as a decimal. This gives the total discount amount. Subtract this amount from the list price to find the final sale price.

What is the formula for the sale price?

The sale price can be found in one step using the formula $\text{Sale Price} = \text{List Price} \times (1 - \text{Discount Rate})$.

What is a marked price?

The marked price is another term for the list price or original price. It is the cost before any discount is applied.