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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

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 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 is12.75.

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*}, of78.75. 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 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 was27.20.

### 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*}, is11.19. 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*}

#### 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 is19.25.

#### 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 is381.50.

#### 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 is53.06.

### 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?

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Color Highlighted Text Notes

### Vocabulary Language: English

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.