Question

1. a fair booth sells: - game tickets for $2.00 each - prize bags for $3.00 each jenna has exactly $24.00 to spend and she must follow these rules: - she buys more game tickets than prize bags. - she buys at least 1 prize bag. - she spends all of the money. a. what are all possible combinations of game tickets and prize bags? b. which combination gives the greatest total number of items? 17. a new bicycle is priced at $350.00. the store first offers a 20% discount off the original price. after the discount is applied, a 5% sales tax on the discounted price is added. what is the final price a customer must pay for the bicycle? 18. a dishwasher was on sale for $480.00. this sale price represented a 20% markdown from the original retail price. after the sale ended, the store raised the price by 10% from the original retail price. what is the new, final price of the dishwasher?

Explanation:

Problem 16a

Step1: Define variables

Let \( x \) be the number of game tickets and \( y \) be the number of prize bags. The cost equation is \( 2x + 3y = 24 \), with \( x > y \) and \( y \geq 1 \).

Step2: Solve for \( x \)

From \( 2x + 3y = 24 \), we get \( x=\frac{24 - 3y}{2}=12-\frac{3y}{2} \).

Step3: Test \( y \) values

• \( y = 1 \): \( x=\frac{24 - 3(1)}{2}=\frac{21}{2}=10.5 \) (not integer, discard).
• \( y = 2 \): \( x=\frac{24 - 3(2)}{2}=\frac{18}{2}=9 \) (9 > 2, valid).
• \( y = 3 \): \( x=\frac{24 - 3(3)}{2}=\frac{15}{2}=7.5 \) (discard).
• \( y = 4 \): \( x=\frac{24 - 3(4)}{2}=\frac{12}{2}=6 \) (6 > 4, valid).
• \( y = 5 \): \( x=\frac{24 - 3(5)}{2}=\frac{9}{2}=4.5 \) (discard).
• \( y = 6 \): \( x=\frac{24 - 3(6)}{2}=\frac{6}{2}=3 \) (3 < 6, discard).
• \( y \geq 7 \): \( 3y \geq 21 \), \( 2x \leq 3 \), \( x < 2 \), but \( x > y \geq 7 \) impossible.

Step1: Calculate total items for each combination

• For (9, 2): \( 9 + 2 = 11 \) items.
• For (6, 4): \( 6 + 4 = 10 \) items.

Step2: Compare totals

11 > 10, so (9, 2) has more items.

Step1: Calculate discount amount

20% of $350 is \( 0.20\times350 = 70 \). Discounted price: \( 350 - 70 = 280 \).

Step2: Calculate sales tax

5% of $280 is \( 0.05\times280 = 14 \). Final price: \( 280 + 14 = 294 \).

Answer:

(9 game tickets, 2 prize bags), (6 game tickets, 4 prize bags)

Problem 16b