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Discounts never fail to grab our attention. Here we will learn how businesses earn profit even after offering discounts. We will also learn how discount and markup are calculated using simple formulas....Read MoreRead Less

The term “discount” refers to reduced prices or an item being sold for a lower price than it would normally be. For example, a dress being sold at 20% of its normal price.

You may come across products that are on sale while shopping. They are less expensive than the regular price of the products. The discount refers to the difference between the original price and the sale price. Discounts are commonly expressed as a percentage of the original price. As a result, they are examples of percent reductions.

The product of the percent discount and the original price is known as the discount. The formula for the discount offered on a product is:

Discount = Percentage discount \( \times \) Original price

The selling price refers to the amount a customer is willing to pay for a product.

The selling price is the difference between the market or the list price and the discount.

Selling price = Original price – Discount

Selling price = Cost to store + Markup

Let us understand how to find the sale price using an example.

The original price of a baseball bat is $30. A 30% discount is included in the sale price. What is the price of the item on sale?

There are two methods to find the sale price.

In method one, we will use the percent equation to find the discount and then we will solve it using the general formula for finding the sale price.

In method two, we will find the price by subtracting the discount from the original price by considering the original price as 100%. Finally, we will solve it to get the sale price.

**Method 1:** First, find the discount. The discount given is 30% of $30.

Discount = Percent discount Original price (writing percent equation)

Discount = 0.3030 (substituting 0.30 for the percent discount and 30 for the original price).

Discount = $9

Sale price = Original price – Discount

Sale price = 30 – 9

Sale price = $21

Therefore, the sale price of the baseball bat is $21.

**Method 2:** First, we find the price after the discount for a particular discount percent. So, the sale price is 100%-30% = 70% of the original price.

Sale price = 70% of $30

Sale price = 0.7030

Sale price = $21

Therefore, the sale price of the baseball bat is $21.

This is the actual price of the item before any increase (markup) or decrease (discount) in its price.

Let us understand how to find the original price by observing the example below.

A toy is on sale for a 15% discount. What was its original price if the sale price is $142?

As we know, the sale price is 100% – 15% = 85% or 85% of the original price.

\( 142=0.85 P \) (writing the equation)

\( \frac{142}{0.85}=\frac{0.85}{0.85}P \) (dividing each side by 0.85)

\( 167.06=P \) (simplified)

Therefore, the original price of the toy was $167.06.

Let us consider an example to understand the comparison of discounts.

Two different stores are selling the same pair of shoes. Which is the most cost-effective option?

**Store 1:** The original price is $58 and there is a 20% discount.

**Store 1:** The original price is $47 and there is a 40% discount.

Use the tape diagrams to plot the discounts and compare the two tape diagrams to find the best option to buy the shoes.

**Tape diagram 1:** Find the discount in store 1, where the regular price is $58 and the discount is 20%.

\( \text{Discount}=\text{Percent discount}\times \text{Original price} \)

\( \text{Discount}=\frac{20}{100}\times 58=0.20\times 58=\$ 11.6 \)

\( \text{Sale price}=\text{Original price}- \text{Discount} \)

\( \text{Sale price}=58-11.6=\$ 46.4 \)

**Tape diagram 2:** Find the discount in store 2, where the regular price is $47 and the discount is 40%.

\( \text{Discount}=\text{Percent discount}\times \text{Original price} \)

\( \text{Discount}=\frac{40}{100}\times 47=0.40\times 47=\$ 18.8 \)

\( \text{Sale price}=\text{Original price}- \text{Discount} \)

\( \text{Sale price}=47-18.8=\$ 28.2 \)

On comparing the two tape diagrams, we can see that store two had more discount, which is 40%, and a lower sale price, which is $28.2 than the discount offered by store one, which is 20%, and a sale price of $46.4 for the same pair of shoes. Hence, purchasing the shoes from store 2 would be a better option.

Stores purchase products at wholesale prices. They then raise the prices of products before selling them to cover their costs and earn a profit. The retail prices are the prices at which stores sell their goods. The markup is the difference between the wholesale price and the retail price. The cost to the store is the wholesale price or the cost paid by the store to purchase that product. This is represented by the formula:

\( \text{Markup}=\text{Selling price}-\text{Cost to store} \)

For example, a store pays $61 for a belt and the markup is 35%. At what price does John buy the belt?

**Solution:**

Tape diagram:

\( \text{Markup}=38\%\times 61 \)

\( \text{Markup}=0.35\times 61 \)

\(\text{Markup} =21.35 \)

\( \text{Selling price}=\text{Cost to store}-\text{Markup} \)

\( \text{Selling price}=61+21.35 \)

\( \text{Selling price}=82.35 \)

Hence, John buys the belt for $82.35.

**Example 1: **You go to a store and buy a pair of earrings for your mother for $115. Later, you come to know that the price was marked up by 15%. At what cost did the store get the earrings?

**Solution: **

Let the cost to store be C.

\( \text{Markup}=15\%\times C \)

\( \text{Markup}=0.15C \)

\( \text{Cost to store}=\text{Selling price}-\text{Markup} \)

\( C=115-0.15C \)

\( C+0.15C=115 \)

\( 1.15C=115 \)

\( C=\frac{115}{1.15}=\$100 \) (approximately)

Hence, the cost to store is $100 (approximately)

**Example 2: **A hammer usually retails at $55 in its original form. There is a 10% discount on it. What is the price of the item on sale?

**Solution:**

First, find the discount. The discount given is 10% of $55.

\( \text{Discount}=\text{Percent discount}-\text{Original price} \) (writing percent equation)

\( \text{Discount}=0.10\times 55 \) (substituting 0.10 for percent discount and 55 for original price)

\( \text{Discount}=\$ 5.5 \) (Multiplied)

After that, find the sale price.

\( \text{Sale price}=\text{Original price}-\text{discount} \)

\( \text{Sale price}=55-5.5 \)

\( \text{Sale price}=\$ 49.5 \)

Therefore, the sale price of the hammer is $49.5.

**Check:** Using a tape diagram.

**Example 3: **The price of a bag that normally costs $42 has been reduced by 10%. The store then deducts an additional 5% from the discounted price. What is the new sale price of the bag?

**Solution:**

Calculate the price after the initial discount.

\( \text{First discount}=\frac{10}{100}\times 42=0.10\times 42=\$ 4.2 \)

\( \text{Sale price after 1st discount}=\text{Original price}-\text{1st discount} \)

\( \text{Sale price after 1st discount}=42-4.2 \)

\( \text{Sale price after 1st discount}=\$37.8 \)

Then, after the second discount, figure out the price.

\( \text{Second discount}=\frac{5}{100}\times 37.8=0.05\times 37.8=\$ 1.89 \)

\( \text{Sale price}=\text{Sale price after 1st discount}-\text{discount} \)

\( \text{Sale price}=37.8-1.89 \)

\( \text{Sale price after 1st discount}=\$35.91 \)

Therefore, the new sale price of the bag is equal to $35.91.

**Example 4: **A bike is on sale for a 25% discount. What was the original price if the sale price is $3200?

**Solution:**

As we know, the sale price is 100% – 25% = 75% or 75% of the original price.

\( 3200=0.75P \) (writing the equation)

\( \frac{3200}{0.75}=\frac{0.75}{0.75} P \) (dividing each side by 0.75)

\( 4266.67=P \) (simplified)

Therefore, the original price of the bike was $4266.67.

Frequently Asked Questions

A markup is an amount added to the cost price of the goods to cover operating costs (operating profit) and profit, whereas a discount is a lowering of the catalog price (or list price).

In most cases, the rate is expressed as a percentage. Multiply the rate by the original price to get the discount. Subtract the discount from the original price to get the sale price.

Selling price = original price – discount