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# Calculating Percentage Discounts

Pritha Mandal, Maria Airth, Matthew Bergstresser
• Author
Pritha Mandal

Pritha is an academic editor and technical writer. She holds a Master's degree in Physics, with specialization in Astrophysics. Pritha has edited hundreds of physical science research papers and has co-authored a paper on radio astronomy.

• Instructor
Maria Airth

Maria has taught University level psychology and mathematics courses for over 20 years. They have a Doctorate in Education from Nova Southeastern University, a Master of Arts in Human Factors Psychology from George Mason University and a Bachelor of Arts in Psychology from Flagler College.

• Expert Contributor
Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Learn how to find discount prices from the discount percentage formula. Further, learn the step-by-step method to find the selling price with some examples. Updated: 09/22/2021

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## How to Calculate a Discount?

A prevalent scene in today's shopping scenario is the presence of countless billboards and banners highlighting sales and discounts. A discount on a certain product implies that the seller is offering a price reduction on the product with respect to the product's original price. The discount amount expressed as a percentage of the original price gives the percentage discount on the product.

To understand how to figure out the discount percentage, consider the following important terms:

• Original price {eq}P {/eq} - This is the price of the concerned product before the discount.
• Discount rate {eq}R\% {/eq} - This is the discount percentage being offered with respect to the original price.
• Selling price {eq}S {/eq} - This is the final price of the concerned product after the discount.

The discount price is determined by multiplying the discount rate with the original price. Subsequently, the final selling price of the discounted product is calculated by subtracting the discount price from the original price of the product.

For instance, consider that the original marked price of a piece of clothing is $150, and the seller is offering a 20% discount on the product. The discount price is then 20% of$150, which amounts to $30. The final selling price of the product is found by subtracting the discount price,$30, from the original price, $150, which gives$120. Similarly, if a shop is offering a 15% discount on a product originally priced at $70, the discount price will be 15% of$70, or $10.5, and the selling price will be$59.5.

## Calculating a Percentage Discount

Joe wants to buy a mystery box that costs $125. The box is on sale for 20% off. How much does the box cost after the discount? #### Step 1: Remember the formula for finding the discount price of an item. Where S = sale price, r = discount percentage rate and p = original price, the discount formula is: • S = p - rp #### Step 2: Convert the percentage rate into a decimal - remove the percent sign and move the decimal two places to the left. If there's no decimal, assume one just to the right of the ones place. Here, as you can see, 20% = 0.20. #### Step 3: Insert values (including decimal value of rate) into the discount formula: • S = 125 - 0.2(125) Remember that when two variables are next to each other, it means that you multiply them. #### Step 4: Solve the equation for S: • S = 125 - 25 = 100 #### Step 5: Format the answer with the currency notation, or in this case, dollars, leaving us with$100.

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## Discount Equation

To better understand how to calculate discounts, consider the discount equation below. If {eq}P {/eq} is the original price of a product and {eq}S {/eq} is the final selling price of the product, the percentage discount or discount rate is given by

{eq}R\%=\frac{(P-S)}{P}\times 100\% {/eq}

Now, if the percentage discount or discount rate {eq}R\% {/eq} and the original price {eq}P {/eq} of a product are known, then the discount price can be expressed as

{eq}D=R\%\times P {/eq}

The above expression of the discount formula shows how to calculate the amount that is to be deducted from the original price of the product. The final selling price {eq}S {/eq} of the product is then

{eq}S=P-D {/eq}

{eq}\Rightarrow S=P-(R\%\times P) {/eq}

Thus, the final selling price of a discounted product can be determined by following the steps below.

• Finding the discount price by multiplying the discount rate with the original price.
• Calculating the selling price by subtracting the obtained discount price from the original price.

As an example, consider the calculation of the selling price of a product originally priced at $200, on which a discount of 12% is applied. The final selling price in the above example can be calculated by the following steps. • Here, {eq}R\%=12\% {/eq} and {eq}P=\$200 {/eq}. So, {eq}D=R\%\times P=12\% \times \$200=\frac{12}{100}\times \$200=\$24 {/eq}. • Thus, {eq}S=P-D=\$200-\$24=\$176 {/eq}

So, the final selling price of the product is {eq}\$176 {/eq} ### Alternate Method While the previous section describes how to find a discount price and subsequently determine the final selling price of a discounted product, there is an alternate method, wherein a slightly modified discount percentage formula can be used to directly calculate the selling price. As per the previously discussed method, the final selling price {eq}S {/eq} of a product with an original price {eq}P {/eq} and discount rate {eq}R\% {/eq} is given by {eq}S=P-(R\%\times P) {/eq} To find the alternate method expression, the above formula needs to be simplified by taking {eq}P {/eq} common, as shown below. {eq}S=P-(R\%\times P)=P(1-R\%) {/eq} {eq}\Rightarrow S=P\left ( 1-\frac{R}{100} \right )=P\left ( \frac{100-R}{100} \right ) {/eq} {eq}\Rightarrow S=(100-R)\%\times P {/eq} Thus, the final selling price can be calculated by subtracting the given discount rate from 100 and multiplying this new rate with the original price of the product. To check the validity of this alternate method of calculating the final selling price, consider the same example as above, where a product originally priced at$200 is to be sold at a discount of 12%. Here, {eq}P=\$200 {/eq} and {eq}R\%=12\% {/eq}. According to the alternate method, the final selling price will be {eq}S=(100-R)\%\times P=(100-12)\%\times \$200=88\% \times \$200 {/eq} {eq}\Rightarrow S=\frac{88}{100}\times \$200=\$176 {/eq} This is the same result as that obtained using the previous method. Thus, the alternate method of calculating the selling price of a discounted product is valid. ## Examples Go through the examples below to practice calculating the final selling price of discounted products. #### Example 1 A product originally priced at {eq}\$25 {/eq} is finally sold at {eq}\$20 {/eq}. What is the discount rate that was offered on the original price? Here, {eq}P=\$25 {/eq} and {eq}S=\$20 {/eq}. Now, the formula for calculating the final selling price from the original price and the discount price is {eq}S=P-D\: \: \Rightarrow \: \: D=P-S {/eq} So, here, the discount price {eq}D=\$25-\$20=\$5 {/eq}

Again, the formula for calculating the discount price from the discount rate is

{eq}D=R\%\times P=\frac{R}{100}\times P {/eq}

{eq}\Rightarrow R=\frac{D}{P}\times 100 {/eq}

So, here

{eq}R=\frac{5}{25}\times 100=20 {/eq}

Thus, the required discount rate here was {eq}20\% {/eq}.

What is the selling price of a dozen apples, which were originally marked at {eq}\$50 {/eq} and were given a discount of {eq}5\% {/eq}? ## Answer Format & Alternative Method Let's first take a look at the answer format. Discount calculations imply dealing with monetary units, thus the answer should always be given in proper currency notation. The type of currency to use should match what's given in the original problem. Here we used dollars, but currency could be in any form. For this example, the correct answer is$100. Additionally, it's appropriate to answer a word problem with a full sentence. Thus, the best answer to this word problem is: ''The mystery box costs $100 after the discount has been applied.'' Now let's take a look at the alternative method. In the original formula, we find out the amount of the discount and remove it from what we would normally pay. There's another way to calculate the discount cost: we could figure out the percentage of the cost that's going to be paid first, and then just multiply that rate by the total cost. In the example, there's a 20% discount, which means that Joe will have to pay for 80% of the original cost (100% - 20% = 80%). The sales price is 80% of the original price, so: • S = 0.8(125) = 100 ## Examples #### Example 1 Mary has learned that the$450 purse she'd been wanting is finally on sale at a 45% discount. How much will the purse cost now?

Step 1: Use the formula S = p - rp

Step 2: Convert rate from percent to decimal: 45% = 0.45

Step 3: Plug values into formula: S = 450 - 0.45(450)

Step 4: Solving: S = 450 - 0.45(450) = 450 - 202.5 = 247.5

Step 5: Add the currency unit, which gives us $247.50. Stated in full, the answer is: ''The purse will cost$247.50 with the discount.''

#### Example 2

This time using the second method:

Using the alternative method, how much does a $5.50 toy cost with a 10% discount? Step 1: If 10% is discounted, that leaves 90% to pay. Step 2: Convert to decimal: 90% = 0.9 Step 3: Plug the variables into the formula: S = 0.9(5.50) Step 4: Solving, we get: 4.95 To unlock this lesson you must be a Study.com Member. Create your account Video Transcript ## Calculating a Percentage Discount Joe wants to buy a mystery box that costs$125. The box is on sale for 20% off. How much does the box cost after the discount?

#### Step 1:

Remember the formula for finding the discount price of an item. Where S = sale price, r = discount percentage rate and p = original price, the discount formula is:

• S = p - rp

#### Step 2:

Convert the percentage rate into a decimal - remove the percent sign and move the decimal two places to the left. If there's no decimal, assume one just to the right of the ones place. Here, as you can see, 20% = 0.20.

#### Step 3:

Insert values (including decimal value of rate) into the discount formula:

• S = 125 - 0.2(125)

Remember that when two variables are next to each other, it means that you multiply them.

#### Step 4:

Solve the equation for S:

• S = 125 - 25 = 100

Format the answer with the currency notation, or in this case, dollars, leaving us with $100. ## Answer Format & Alternative Method Let's first take a look at the answer format. Discount calculations imply dealing with monetary units, thus the answer should always be given in proper currency notation. The type of currency to use should match what's given in the original problem. Here we used dollars, but currency could be in any form. For this example, the correct answer is$100. Additionally, it's appropriate to answer a word problem with a full sentence. Thus, the best answer to this word problem is: ''The mystery box costs $100 after the discount has been applied.'' Now let's take a look at the alternative method. In the original formula, we find out the amount of the discount and remove it from what we would normally pay. There's another way to calculate the discount cost: we could figure out the percentage of the cost that's going to be paid first, and then just multiply that rate by the total cost. In the example, there's a 20% discount, which means that Joe will have to pay for 80% of the original cost (100% - 20% = 80%). The sales price is 80% of the original price, so: • S = 0.8(125) = 100 ## Examples #### Example 1 Mary has learned that the$450 purse she'd been wanting is finally on sale at a 45% discount. How much will the purse cost now?

Step 1: Use the formula S = p - rp

Step 2: Convert rate from percent to decimal: 45% = 0.45

Step 3: Plug values into formula: S = 450 - 0.45(450)

Step 4: Solving: S = 450 - 0.45(450) = 450 - 202.5 = 247.5

Step 5: Add the currency unit, which gives us $247.50. Stated in full, the answer is: ''The purse will cost$247.50 with the discount.''

#### Example 2

This time using the second method:

Using the alternative method, how much does a $5.50 toy cost with a 10% discount? Step 1: If 10% is discounted, that leaves 90% to pay. Step 2: Convert to decimal: 90% = 0.9 Step 3: Plug the variables into the formula: S = 0.9(5.50) Step 4: Solving, we get: 4.95 To unlock this lesson you must be a Study.com Member. Create your account • Activities • FAQs ## Percentages Percentages are essentially fractions that have been multiplied by 100. Percentages are useful when expressing discounts. It is helpful to know how to calculate percentage discounts when shopping. Let's look at some practice problems dealing with percentages. You may use a calculator to complete the problems. Show your work. ### Practice 1. A designer shirt is on sale for 30% off its normal price of$20. What is the amount saved and sale price of the shirt?
2. A chair is on sale for $45. Normally it is$75 dollars. What is the percentage discount?
3. A box of chocolate chip cookies is on sale for $3.25. The percentage discount was 15%. What was the original price of the box of cookies? ### Solutions 1. The sale price is$14.

2.

This is a 40% discount.

3.

This value was rounded from $3.7375. We know the original price was the$3.25 hence the 1 in the 1.15 and we have to add in the 15/100 hence the .15 in 1.15. This gives us the original cost of around $3.74. We only round to two decimal places because the lowest coin value is one cent, or$0.01.

#### How do you take 20% off a price?

To take 20% off a price, say $X, the following expression can be used. The final price after taking 20% off$X will be given by

Final price=(100-20)% *X=80% *X=(80/100)*X

#### How do I calculate a discount percentage?

To calculate the discount percentage, first, the discount price needs to be determined. The discount price is equal to the difference between the original price and the final selling price. Then, the discount percentage can be found by dividing the discount price by the original price and multiplying the result by 100.

#### What is the formula of discount%?

The formula of discount % is expressed in terms of the original price P and final selling price S of the concerned product. It is given by the following:

Discount%=(P-S)/P*100%

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