Question

Becky and Michele are both shopping for a new car at two different dealerships. Dealership A is offering $500 cashback on any purchase, while dealership B
is offering $1,000 cash back. The tax rate is 5% at dealership A but 8% at Dealership B. Becky wants to buy a car that is $15,000, and Michele is planning to buy a car that costs $20,000. Use algebraic expressions to help you answer the following question.

1. At which dealership will Becky get the better deal? how much does she save?

2. At which dealership will Michele get the better deal? how much does she save?

3. What gerneralization can you make that would help any shopper know which dealership has the better deal? At which price point would the two deals be equal.

Answer 1

Dealership A : 500 cash back , 5% tax
Becky : 15,000 + 0.05(15000) - 500 = 15,000 + 750 - 500 = 15,250
Michele : 20,000 + 0.05(20,000) - 500 = 20,000 + 1000 - 500 = 20500

dealership B : 1000 cash back, 8% tax
Becky : 15,000 + 0.08(15,000) - 1000 = 15000 + 1200 - 1000 = 15,200
Michele : 20,000 + 0.08(20,000) - 1000 = 20,000 + 1600 - 1000 = 20600

1. Becky will get the better deal at dealership B.....she saves 50 bucks
2. Michele will get the better deal at dealership A...she saves 100 bucks
3. ?....sorry

Answer 2

A. First write an algebraic expression to represent each dealership’s special. Dealership A p + 0.05p - 500 = 1.05p - 500 Dealership B p + 0.08p - 1000 = 1.08p - 1000 Now substitute the price that Becky plans to pay to see which deal is better for her. Dealership A: 1.05(15,000) - 500 = 15,750 - 500 = 15,250 Dealership B: 1.08(15,000) - 1000 = 16,200 - 1000 = 15,200 Dealership A > Dealership B So, Becky will get a better deal with Dealership B, since it is $50 cheaper. b. Substitute the price that Michele plans to pay into each expression. Dealership A: 1.05(20,000) - 500 = 21,000 - 500 = 20,500 Dealership B: 1.08(20,000) - 1000 = 21,600 - 1000 = 20,600 Dealership A < Dealership B So, Michele will get a better deal with Dealership A, since it is $100 cheaper. c. The two deals are equivalent when the additional 3% tax equals $500 (0.03p = 500), which occurs when the purchase price is $16,667. Students’ generalizations may suggest that any shopper planning to buy a car with a price less than $16,667 will get the better deal at Dealership B, while anyone planning to buy a car at a price higher than $16,667 will get a better deal at Dealership A.