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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%