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Prices Involving Discounts

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Find Discount Prices Given Sales
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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: 29.99 \times .60 = 17.99

Subtract discount from original price: 29.99 - 17.99 = 12.00

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: 12 \times .0625 = .75

Add tax to the sale price: 12 + .75 = 12.75

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:

\text{original price} + 5\% \ \text{of original price} = \text{total cost}

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

p+.05p = c

In this equation, we p and .05p are like terms so they can be combined.

1.05p = c

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

1.05p &= 78.75\\1.05p \div 1.05 &= 78.75 \div 1.05\\p &= 75

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.

1.07p &= 29.10\\p &= 27.20

The original price of the item was $27.20.

Find each new price given a discount of 30%.

Example A

$27.50

Solution: $19.25

Example B

$545.00

Solution: $381.50

Example C

$75.80

Solution: $53.06

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: 15.99 \times .30 = 4.80

Subtract discount from original price: 15.99 - 4.80 = 11.19

The sale price is $11.19.

Don’t forget the sales tax!

Amount of tax: 11.19 \times .05 = .56

Add tax to the sale price: 11.19 + .56 = 11.75

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: 55.00 \cdot 2 = 110.00

2.50 \cdot 6 = 15.00

Total price of products: 125.00

Calculate the tax of 5.75%: 125 \times .0575 = 7.19

Add the tax to the price: 125.00 + 7.19 = 132.19

Calculate the 10% discount: 132.19 \times .10 = 13.22

Subtract the discount from the price: 132.19 - 13.22 = 118.97

The grand total is $118.97.

Video Review

Finding Discounts

Practice

Directions: Your food bill at a restaurant is $85.77, calculate your total cost after the tip. 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%

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

  1. total cost: $1475.68 tax rate: 7%
  2. total cost: $63.80 tax rate: 4.5%
  3. total cost: $55.90 tax rate: 4%
  4. total cost: $80.20 tax rate: 2.5%
  5. total cost: $120.00 tax rate: 5%
  6. total cost: $99.50 tax rate: 2%
  7. total cost: $155.30 tax rate: 3%
  8. 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.

  1. 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?
  2. 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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