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# Discount|Definition & Meaning

## Definition

The **reduction** in **original price** of a product or service under **special conditions** or otherwise is called **discount.** It may be an absolute value, a **percentage value** or a cashback offer.

The term **discount** is very common and we are all familiar with the **concept** when it comes to the shopping perspective. **Mathematically,** its just a reduction in price specified in the form of some **absolute cash price** or in terms of **percentage. **Following **figure** represents a photo shows a **sample discount offer** typically published by **vendors** and stores all over the world.

**Figure 1: Introduction to Discount**

## Explanation of Discount

In **mathematics, discount** is a percentage or fractional value that characterizes **reduction** in the price of a **product** or **service.** Consider the example of a **bicycle.** Lets say that the list price of the bicycle was **800** united states dollars. Now the store owner wants to increase the **cash flow** to accommodate new line of bicycles. To achieve this he or she offers a **discount** of **twenty percent.**

**Figure 2: Bicycle Example of Discount**

The **twenty percent** of the list price of the bicycle is **800 x 20 / 100 = 160**. Hence, the new price after the deduction of **discount,** can easily be calculated by using simple subtraction **800 – 160 = 640** united states dollars.

## Mathematical Formula for Discount

In the above example **new price** and the **discount** was calculated using some **intuition.** In this section, we formally introduce the **mathematical form** of the discount calculations. Lets assume that the **original price** of a given product or service is **symbolized by ‘O’.** The **new price** of the same product or service after discount is **symbolized by ‘N’**. And the **discount rate** in percentage terms is **symbolized by ‘d’.**

**Figure 3: Mathematical Form of Discount and New Price**

Under these **abstractions,** we can write the **following formula:**

N = O$\mathsf{\left(1-\dfrac{d}{100}\right)}$

The **discounted** or saved amount **‘D’** can be calculated by using the following **formula:**

Discount Amount = D = O$\mathsf{\left(\dfrac{ d }{ 100 }\right)}$

## Applications of Discount

There are many **examples** of **discount** that we may see in daily life where we may have to **calculate** the **new price** after the **discounted amount.** We may have to make decisions about whether or not to buy a **certain product.** You may have to compare two or more products with each other while **adjusting** the **discounts** for some or all of them.

Suppose you wanted to **buy** **a** **house** for the family. Last week you checked a beautiful house worth **eighty thousand** united states dollars. At that time you couldn’t afford it since your budget was **seventy thousand** united states dollars. Now you receive an offer from your real estate dealer that the house owner is willing to give a **15 percent discount** if the buyer pays in cash.

To see whether or not the house is in **your range,** you may use the **formulae** given above to find the **new price** of the house as follows:

N = \$80,000$\mathsf{\left(1-\dfrac{ 15 }{ 100 }\right)}$ = \$68,000

Now that you know that the new price would be **sixty eight thousand** united states dollars which is within your budget of **seventy thousand,** you can easily make a decision and confirm you order to the dealer.

**Figure 4: Example of Discount on a New House**

## Other Uses Cases of the Term Discount

The **term discount** may have **different meaning** in **different areas** of mathematics. Following are some of the use cases in the field of **business mathematics** or the mathematics of economics.

### Discount Loan

In any **loan agreement,** the lender and borrower reach an agreement to how the **interest amount** and the **principle amount** will be paid over the whole tenure. Sometimes, the lender decides to deduct the **interest amount** in first few **installments** while maintaining the principle amount at a stand still. The type of loan where the **interest payment** is deducted mostly in **first few installments** is called **discount loan.**

### Discount Rate

The **discount rate** is a term often used synonymous to the term **interest rate.** More formally, the discount rate is the interest rate **declared by** the **central bank** or any central authority responsible for making the **fiscal policy** of a country. This **discount rate** is used as a reference **interest rate** for rolling out discount offers and money **borrowing packages** of the **commercial banks.**

## Numerical Examples of Discounts

### Example 1

The **listed price** of a **sports car** is **ninety thousand** dollars. You have heard that the car manufacturer has announced a **discount of 25%** for the 100% upfront payment on **Christmas.** What would be the **discount amount** and the **new price** of the car?

### Solution

**Given:**

d = 25%

O = 90,000 USD

**Calculating the discount amount:**

D = O $\mathsf{\left( \dfrac{ d }{ 100 }\right)}$

D = 90,000 USD $\mathsf{\left(\dfrac{ 25 }{ 100 } \right)}$

D = 90,000 USD ( 0.25 )

D = 22,500 USD

**Calculating the new price:**

N = O $\mathsf{\left(1-\dfrac{ d }{ 100 }\right)}$

N = 90,000 USD $\mathsf{\left(1-\dfrac{ 25 }{ 100 } \right)}$

N = 90,000 USD ( 0.75 )

N = 67,500 USD

### Example 2

Suppose that you want to **visit Turkey** for the term break. The price of the **seven day tour** of a single individual is **two thousand dollars.** The travel company has a package for group tours. Under this package, if you book the travel plan as a **group of four people,** you get a discount of **fifteen percent.** What will be the value of **discount amount** in dollars. What will be the **revised rate** after the discount for a single individual.

### Solution

**Given:**

d = 15 %

O = 2,000 USD

**Calculating the discount amount:**

D = O $\mathsf{\left(\dfrac{ d }{ 100 }\right)}$

D = 2,000 USD $\mathsf{\left(\dfrac{ 15 }{ 100 }\right)}$

D = 2,000 USD ( 0.15 )

D = 300 USD

**Calculating the new price:**

N = O $\mathsf{\left(1-\dfrac{ d }{ 100 }\right)}$

N = 2,000 USD $\mathsf{\left(1-\dfrac{ 15 }{ 100 }\right)}$

N = 2,000 USD ( 0.85 )

N = 1,700 USD

*All images were created with GeoGebra.*