An arcade offers customers the option to put money on pre-paid cards and then swipe their cards to pay for each game played. Games also award customers with tickets dispersed at regular intervals while playing each game.

Question

The equation 𝑨(𝒈) = 𝟏𝟓 − 𝟎. 𝟓𝟎𝒈 represents the amount of money remaining, 𝑨, on a customer’s card after playing 𝒈 games. Which statement is true?

GPT-4o mini · Answer

To analyze the equation $ A(g) = 15 - 0.50g $, let's break down its components: - $ A(g) $ represents the amount of money remaining on the card after playing $ g $ games. - The initial amount on the card is $15. - For each game played, the customer spends $0.50 (since playing $ g $ games costs $ 0.50g $). We can derive several insights from this equation: 1. **Initial Amount:** When $ g = 0 $ (no games played), the amount remaining on the card is $ A(0) = 15 - 0.50(0) = 15 $. So, the initial amount on the card is $15. 2. **Cost per Game:** Each game played reduces the amount of money on the card by $0.50. Therefore, the total cost for $ g $ games is $ 0.50g $. 3. **Limit:** The function $ A(g) $ represents a linear decrease in the amount of money on the card. If the customer plays enough games (specifically when $ A(g) $ reaches $0), they will run out of money on the card. To find out how many games that would take, we can set the equation to zero: $$ 15 - 0.50g = 0 $$ Solving for $ g $: $$ 0.50g = 15 \ g = \frac{15}{0.50} = 30 $$ This means the customer can play a maximum of 30 games before the card balance reaches $0. 4. **Interpretation of $ A(g) $:** The equation shows that the amount remaining on the card decreases linearly with the number of games played, confirming the expenditure of money per game. Based on this analysis, **the true statement** regarding this equation is: **The customer can play 30 games before their card balance reaches $0.**