Renata wins a $20 gift card to an online music site. After Renata purchases 16 songs, the gift balance is $0. Which equation represents the relationship between $y$, the remaining balance, and $x$, the number of songs purchased?
$4 x+5 y=-100$
$5 x+4 y=80$
$4 x+5 y=16$
$5 x+4 y=20$
Answer (1)
Calculate the cost per song: 16 20 = 1.25 .
Express the remaining balance y in terms of the number of songs purchased x : y = 20 − 1.25 x .
Rewrite the equation using fractions: y = 20 − 4 5 x .
Rearrange the equation to match the given options: 5 x + 4 y = 80 , so the final answer is 5 x + 4 y = 80 .
Explanation
Understanding the Problem Renata starts with a gift card worth $20 . After buying 16 songs, the card's balance is $0 . We want to find the equation that connects y (the remaining balance) with x (the number of songs bought).
Calculating the Cost Per Song First, we need to determine the cost per song. Since the $20 gift card covers 16 songs, we can calculate the cost per song as follows: Cost per song = Number of songs Total value of gift card = 16 20 = 1.25 So, each song costs $1.25 .
Formulating the Equation Now, we can express the remaining balance y in terms of the number of songs purchased x . Renata starts with $20 , and each song reduces the balance by $1.25 . Therefore, the equation is: y = 20 − 1.25 x We can also write 1.25 as 4 5 , so the equation becomes: y = 20 − 4 5 x
Rearranging the Equation To match one of the given options, let's rearrange the equation. Multiply both sides by 4: 4 y = 80 − 5 x Now, add 5 x to both sides: 5 x + 4 y = 80 This matches the second option.
Final Answer Therefore, the equation that represents the relationship between the remaining balance y and the number of songs purchased x is: 5 x + 4 y = 80
Examples
Imagine you're managing a prepaid card for online purchases. This problem helps you track how much money is left on the card after each purchase. Understanding this linear relationship allows you to predict when you'll need to reload the card, ensuring you don't run out of funds unexpectedly. This is useful for budgeting, managing expenses, and making sure you stay within your spending limits.
