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Prices Involving Discounts

Determine the value of a number that has been reduced by a given percentage.

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Find Discount Prices Given Sales
License: CC BY-NC 3.0

Jeremiah works at a local bookstore and convinced his boss to have a 30% off everything sale. The sales tax is 5%. What is the total price on a book whose cover price is $15.99?

In this concept, you will learn to find discount prices given sales.

Sale Prices

In order to reduce inventories or motivate buyers, stores often have sales in which they discount prices by a percent. Instead of a percent increase, this is a percent decrease, which results in a sale price.

Let’s look at an example.

At the end of the summer, a clothing store puts all swimwear on sale. They offer a discount of 60%. If the regular price is $29.99 on a bikini, what is the sale price?

First, multiply the original price by the percent decrease. Remember that percent means the denominator is 100.

\begin{align*}\begin{array}{rcl} 29.99 \times \frac{60}{100} &=& \frac{1799.4}{100} \\ &=& 17.994 \end{array}\end{align*}Next, subtract the amount of decrease from the original price.

\begin{align*}29.99 - 17.994 = 11.996 \end{align*}

The answer is 11.996.

The sale price of the bikini is $12.00.

If the sales tax is 6.25%, what is the final price of the discounted bikini?

First, multiply the sale price by the sales tax. Remember that percent means the denominator is 100.

\begin{align*}\begin{array}{rcl} 12.00 \times \frac{6.25}{100} &=& \frac{75}{100} \\ &=& 0.75 \end{array}\end{align*}

Next, add the sales tax to the sale price.

\begin{align*}12.00 + 0.75 = 12.75\end{align*}

The answer is 12.75.

The total price of the bikini is $12.75.

Calculating a price with tax is the same as increasing by a percent. What if you know the total price including the tax and want to know the original price of a product?

Let’s look at an example.

A store clerk charges you $78.75 for a DVD player. The tax in your area is 5%. So how much was the original price of the DVD player?

First, in order to get the total cost, the cash register computes the 5% tax and adds the tax to the original price. In other words: 

\begin{align*}\text{Original Price} + 5 \% \ \text{of Original Price} = \text{Total Cost}\end{align*}

Let \begin{align*}p\end{align*} represent the original price and \begin{align*}c\end{align*} represent the total cost, the equation becomes:

 \begin{align*}p + 0.05 \ p= c \qquad \ \text{or}\qquad \ 1.05 \ p = c\end{align*}

In this case, if you are given \begin{align*}p\end{align*}, you can solve for \begin{align*}c\end{align*}. Or if you are given \begin{align*}c\end{align*}, you can solve for \begin{align*}p\end{align*}.

Next, fill in what you know. In this example, you paid a total cost, \begin{align*}c\end{align*}, of $78.75. Substitute \begin{align*}c\end{align*} in the equation.

\begin{align*}\begin{array}{rcl} 1.05 \ p &=& c \\ 1.05 \ p &=& 78.75 \end{array}\end{align*}

Then, divide both sides by 1.05 to solve for \begin{align*}p\end{align*}.

\begin{align*}\begin{array}{rcl} 1.05 \ p &=& 78.75 \\ \frac{1.05 \ p}{1.05} &=& \frac{78.75 }{1.05} \\ p &=& 75 \end{array}\end{align*}

The answer is 75.

The original price was $75.00

Let’s try another one.

You are charged $29.10 for an item with 7% tax included. What was the original price of the item?

First, write an equation.

\begin{align*}p + 0.07 \ p = c \qquad \text{or} \qquad 1.07 \ p = c\end{align*}

Next, fill in what you know. In this example, you paid a total cost, \begin{align*}c\end{align*}, of $29.10. Substitute \begin{align*}c\end{align*} in the equation.

\begin{align*}\begin{array}{rcl} 1.07 \ p &=& c \\ 1.07 \ p &=& 29.10 \end{array}\end{align*}

Then, divide both sides by 1.07 to solve for \begin{align*}p\end{align*}.

\begin{align*}\begin{array}{rcl} 1.07 \ p &=& 29.10 \\ \frac{1.07 \ p}{1.07} &=& \frac{29.10 }{1.07} \\ p &=& 27.196 \end{array}\end{align*}The answer is 27.196.

The original price was $27.20.

Examples

Example 1

Earlier, you were given a problem about Jeremiah’s bookstore sale.

They have a 30% off any book sale. You bought a book for $15.99 and then had to pay 5% sales tax on the book purchase.

First, calculate the discount on the purchase.

\begin{align*}\begin{array}{rcl} 15.99 \times \frac{30}{100} &=& \frac{479.7}{100} \\ &=& 4.797 \end{array}\end{align*}

Next subtract the discount from the original price to find the price of your book.

\begin{align*}15.99 - 4.797 = 11.193\end{align*}

Then, calculate the sales tax. The price of the book, \begin{align*}p\end{align*}, is $11.19. You are looking for the cost, \begin{align*}c\end{align*}.

\begin{align*}\begin{array}{rcl} c &=& 1.05 \ p \\ c &=& 1.05 (11.19) \\ c &=& 11.7495 \\ \end{array}\end{align*}

The answer is 11.7495.

The final price is $11.75.

Example 2

Brenda purchases some plants for her gardens. Two trees have a price of $55.00 each. Six tulips cost $2.50 each. The tax on Brenda’s purchase is 5.75% but there is an early-bird special of 10% off the entire purchase to those who show up before 10 am (which Brenda did). What is Brenda’s total cost?

First, add up all of your purchases before tax and discounts.

\begin{align*}2 \ \text{trees} + 6 \ \text {tulips} \\\end{align*}

\begin{align*}2\times 55 + 6 \times 2.50 = 125\end{align*}

Next, calculate the total cost after sales tax.

\begin{align*}\begin{array}{rcl} c &=& 1.0575 \ p \\ c &=& 1.0575 \times 125 \\ c &=& 132.1875 \\ \end{array}\end{align*}

Next, calculate the discount on the purchase.

\begin{align*}\begin{array}{rcl} 132.1875 \times \frac{10}{100} &=& \frac{1321.875} {100} \\ &=&13.21875 \end{array}\end{align*}Then, subtract the discount from the total price to find the price paid.

\begin{align*}132.1875 - 13.21875 = 118.96875\end{align*}

The answer is 118.96875.

The final price Brenda paid for the trees and tulips is $118.97.

Example 3

The original price is $27.50. Find the new price given a discount of 30%.

First, calculate the discount on the purchase.

\begin{align*}\begin{array}{rcl} 27.50 \times \frac{30}{100} &=& \frac{825}{100} \\ &=& 8.25 \end{array}\end{align*}

Next, subtract the discount from the original price to find the price paid.

\begin{align*}27.50 - 8.25 = 19.25\end{align*}

The answer is 19.25.

The final price is $19.25.

Example 4

The original price is $545.00. Find the new price given a discount of 30%.

First, calculate the discount on the purchase.

\begin{align*}\begin{array}{rcl} 545.00 \times \frac{30}{100} &=& \frac{16350}{100} \\ &=& 163.5 \end{array}\end{align*}

Next, subtract the discount from the original price to find the price paid.

\begin{align*}545.00 -163.50 = 381.50\end{align*}

The answer is 381.50.

The final price is $381.50.

Example 5

The original price is $75.80. Find the new price given a discount of 30%.

First, calculate the discount on the purchase.

\begin{align*}\begin{array}{rcl} 75.80 \times \frac{30}{100} &=& \frac{2274}{100} \\ &=& 22.74 \end{array}\end{align*}

Next, subtract the discount from the original price to find the price paid.

 \begin{align*}75.80 - 22.74 = 53.06\end{align*}

The answer is 53.06.

The final price is $53.06.

Review

Your food bill at a restaurant is $85.77. Calculate your total cost after each of the following. Be sure to round when necessary.

1. Service 10%

2. Decent service 15%

3. Great service 20%

4. Outstanding service 25%

5. Poor service 5%

What was the original price given total cost and tax rate? Be sure to round when necessary.

6. Total cost: $1475.68 tax rate: 7%

7. Total cost: $63.80 tax rate: 4.5%

8. Total cost: $55.90 tax rate: 4%

9. Total cost: $80.20 tax rate: 2.5%

10. Total cost: $120.00 tax rate: 5%

11. Total cost: $99.50 tax rate: 2%

12. Total cost: $155.30 tax rate: 3%

13. Total cost: $250.75 tax rate: 3.5%

Solve each problem. There are several steps to solving each problem. Be sure to round when necessary.

14. You take a taxi ride in a foreign country where they add 20% to your total for late night travel. The driver expects an additional 15% tip. How much do you owe for the taxi ride if the meter shows $45?

15. You take your mother out for lobster for Mother’s Day. The lobster platters are $24.95 each but include the drink and dessert buffet. Your waitress is a mother, too, so you leave her a 20% tip. However, you did bring a coupon for 25% off. What is your total cost for 2 people?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.14. 

Resources

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Vocabulary

Markup

The markup is the difference between the wholesale price and the retail price.

Wholesale price

The wholesale price is the price that a merchant pays when they purchase a product from a manufacturer.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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