# 5.14: Find Discount Prices Given Sales

**At Grade**Created by: CK-12

**Practice**Prices Involving Discounts

Have you ever purchased a book from a bookstore? Take a look at this dilemma.

A local bookstore has a 30% off everything sale. The sales tax is 5%. What is the total price on a book whose cover price is $15.99?

**Use this Concept to learn how to solve this problem. You will see it again at the end of the Concept.**

### Guidance

Fortunately for the consumer, price changes aren’t always increases. 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. We can calculate these *sale prices.*

Let's take a look at how to solve these problems.

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?

**Amount of discount:** \begin{align*}29.99 \times .60 = 17.99\end{align*}

**Subtract discount from original price:** \begin{align*}29.99 - 17.99 = 12.00\end{align*}

**The sale price is only $12.00.**

**Not so fast. We almost forgot to pay the tax. If the sales tax is 6.25%, we must now add the tax to the sale price.**

**Amount of tax:** \begin{align*}12 \times .0625 = .75\end{align*}

**Add tax to the sale price:** \begin{align*}12 + .75 = 12.75\end{align*}

**The sale price with tax is $12.75.**

Here is another situation.

*Notice that with a discount you subtract the amount and with sales tax you add the cost.*

**Absolutely! A lot of this math you are probably able to figure out in your head. However, it is a good idea to understand the steps and how the pricing is figured out!**

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?**

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?

**We can find this out by using an equation.**

In order to get the total cost, the cash register computes the 5% tax and adds the tax to the original price like we did three sections ago. In other words:

\begin{align*}\text{original price} + 5\% \ \text{of original price} = \text{total cost}\end{align*}

Now let’s use variables and convert the percent to decimal.

\begin{align*}p+.05p = c\end{align*}

In this equation, we \begin{align*}p\end{align*} and \begin{align*}.05p\end{align*} are like terms so they can be combined.

\begin{align*}1.05p = c\end{align*}

In this case, if we are given \begin{align*}p\end{align*}, we can solve for \begin{align*}c\end{align*}. Or if we are given \begin{align*}c\end{align*}, we can solve for \begin{align*}p\end{align*}. In our example, we paid a total cost \begin{align*}c\end{align*} of $78.75. Substitute \begin{align*}c\end{align*} in the equation and solve for \begin{align*}p\end{align*}.

\begin{align*}1.05p &= 78.75\\ 1.05p \div 1.05 &= 78.75 \div 1.05\\ p &= 75\end{align*}

**The original price was $75.00.**

Here is another one.

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

First, let's write an equation and solve it.

\begin{align*}1.07p &= 29.10\\ p &= 27.20\end{align*}

**The original price of the item was $27.20.**

Find each new price given a discount of 30%.

#### Example A

\begin{align*}$27.50\end{align*}

**Solution: \begin{align*}$19.25\end{align*}**

#### Example B

\begin{align*}$545.00\end{align*}

**Solution: \begin{align*}$381.50\end{align*}**

#### Example C

\begin{align*}$75.80\end{align*}

**Solution: \begin{align*}$53.06\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

Let's go through each step of the solution.

Amount of discount: \begin{align*}15.99 \times .30 = 4.80\end{align*}

Subtract discount from original price: \begin{align*}15.99 - 4.80 = 11.19\end{align*}

The sale price is $11.19.

**Don’t forget the sales tax!**

Amount of tax: \begin{align*}11.19 \times .05 = .56\end{align*}

Add tax to the sale price: \begin{align*}11.19 + .56 = 11.75\end{align*}

**The sale price with tax is $11.75.**

### Vocabulary

- Wholesale price
- the price that a merchant pays when they purchase a product from a manufacturer.

- Markup
- the amount that the merchant charges retail to the customer. The difference between wholesale and retail is the markup and also the profit.

- Sales Tax
- a percentage charged on purchases, and that goes to the government.

- Discounts
- when merchants have a sale a percent of the decrease is calculated to sell the product at a lower price.

### Guided Practice

Here is one for you to try on your own.

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

**Solution**

**Add up all of the products:** \begin{align*}55.00 \cdot 2 = 110.00\end{align*}

\begin{align*}2.50 \cdot 6 = 15.00\end{align*}

**Total price of products:** 125.00

**Calculate the tax of 5.75%:** \begin{align*}125 \times .0575 = 7.19\end{align*}

**Add the tax to the price:** \begin{align*}125.00 + 7.19 = 132.19\end{align*}

**Calculate the 10% discount:** \begin{align*}132.19 \times .10 = 13.22\end{align*}

**Subtract the discount from the price: \begin{align*}132.19 - 13.22 = 118.97\end{align*}**

**The grand total is $118.97.**

### Video Review

### Practice

Directions: Your food bill at a restaurant is $85.77, calculate your total cost after the tip. Be sure to round when necessary.

- service 10%
- decent service 15%
- great service 20%
- outstanding service 25%
- poor service 5%

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

- total cost: $1475.68 tax rate: 7%
- total cost: $63.80 tax rate: 4.5%
- total cost: $55.90 tax rate: 4%
- total cost: $80.20 tax rate: 2.5%
- total cost: $120.00 tax rate: 5%
- total cost: $99.50 tax rate: 2%
- total cost: $155.30 tax rate: 3%
- total cost: $250.75 tax rate: 3.5%

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

- 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?
- 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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### Image Attributions

Here you'll find discount prices given sales.