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Applications of Proportional Relationships

- TLW use proportions to find taxes and gratuities, markups and markdowns on items for sale, commission and fees, simple interest, percent increase or decrease, and percent error.

Proportion

- Proportions can be used to solve problems involving percents. The following proportion can be used to find an unknown value.

Sales Tax and Gratuities

- Percents are often used when describing and solving problems involving sales tax and gratuities.
- The sales tax rate varies by state, and sometimes by city. It is a percent of the total cost of items purchased.

Sales Tax Example

- Larry bought a video console for $129. The sales tax in the state is 6%. How much tax did Larry pay for the video game console?
- Set up a proportion , and then cross multiply. Let n represent the amount of sales tax .
100(n) = 129(6) Larry paid $7.74 in sales tax

100n = 774

N = 7.74

Gratuity

- A gratuity is a tip, or bonus, that is given for services that have been provided. It is usually computed as a percent of the total cost for a service.

Gratuity Example

Kyle’s family went out to dinner. The total for the bill was $45. Kyle’s dad left a 15% tip. What was the amount of the tip?

Set up a proportion

100n = 675

N = 6.75

Tip was $6.75

Markups

- A markup is the difference between the cost of an item and its selling price.
- If you know the cost of an item and the markup, you can find the selling price.
- Selling price = cost + markup

Markup Example

- A store buys sweatshirts for $ 12 each and the marks up the price by 25%. What is the price of the sweatshirt at this store?
- Set up a proportion
100n = 300

Mark up is $ 3 n= 3

Total cost is 12 + 3 = $15

Markdowns

- A markdown is the amount of an item reduced from its regular price.
- If you know the original price and the markdown, you can find the sale price.
- Sale price = original price – markdown

Markdown Example

- The original price of a pair of shoes was $49. The price of the shoes are marked down 30%. What is the sale price of the shoes?
- Set up proportion
- n⁄49 = 30⁄100
- 100n = 1470 49.00 – 14.70 = 34.30
- N= 14.70 Sale price is: $ 34.30
- Markdown is $14.70

Commissions and Fees

- A commission is an amount of money paid to a salesperson based on his or her total amount of sales and the commission rate.
- A fee is an amount added to the cost of an item for service.

Commission Example

- Richard is paid 8% commission on his carpet sales. How much commission will Richard earn for sales of $1650?
- Set up proportion
- n⁄1650 = 8⁄100
- 100n = 13,200
- N= 132
- Richard will earn $132 in commission

Fee Example

- Mindy bought a gift online for $15. This included a shipping fee of $3. What percent of the cost of the gift was the shipping fee?
- Set up proportion
- 3⁄15 = n⁄100
- 15n = 300 Shipping fee was 20%
- N =20
- 20/100 = 20%

Interest Rate

- Interest is the amount you are charged when you borrow money or the amount you are paid when you invest money.
- The principal is the original amount invested or borrowed.
- The interest rate is a percent of the principal.

Simple Interest

- Simple interest is paid only on the principal and is paid at the end of an investment time period.
- TO FIND SIMPLE INTEREST USE THE FORMULA:
I = prt

I= amount of Interest

P= principal

R= interst rate

T= time in years

Simple interest rate example

- Tom borrowed $ 300 to buy a mountain bike. If he pays interest at a rate of 4% on a 6-month loan, how much interest will he pay?
- I=prt
- 300 ( 4%) ( ½)
- 300(.04)(.5)
- = 6
- Tom will pay $6 interest on his loan

Simple interest example # 2

- Vic deposited $2500 into a savings account that pays her simple interest. The interest rate is 3%. How much interest will Vic earn in 2 years? What will Vic’s account balance be after 2 years?
- I=prt
- = 2500(3%)(2)
- =2500(.03)(2)
- =150
- 2500+150= $2650

Percent of Increase or Decrease

- The percent of increase or decrease is the change from a given amount expressed as a percent of that amount.
- To find the percent of increase use the following proportion:
- New amount- original amount/ original amount =n/100
- To find the percent of decrease use the following proportion:
- Original amount – new amount / original amount =n/100

Percent of Increase Example

- In one year, Braden’s height went from 60 inches to 63 inches. What was the percent of increase in his height?
- New amount- original amount/ original amount =n/100
63-60/60 = n/100

3/60 =n/100

300 = 60n

N=5

5/100 = 5%

His height increased 5% in one year.

Percent of decrease Example

- Tara purchased a DVD player for $150. Six months , the same DVD player was selling $120. What was the percent of decrease on the price?
- Original amount – new amount / original amount =n/100
- 150-120/150 = n/100
- 30/150 = n/100
- 3000=150n 20=n
- 20/100= 20%
- DVD player decreased by 20%

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