Question

Tavon has a gist card for $85 that losses $3.50 for each 30-day period is not used. He has another gift card for $75 that losses $3 for each 30-day period it is not used. A. Write and solve an equation for the number of 30-day periods until the value of the gift cards will be equal. B. What will the value of each card be when they have equal value?

Answer 1

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Explanation:

Let's denote the initial values of the gift cards as follows:

- Initial value of the $85 gift card: \( G_1 = 85 \)

- Loss per 30-day period for the $85 gift card: \( L_1 = 3.50 \)

- Initial value of the $75 gift card: \( G_2 = 75 \)

- Loss per 30-day period for the $75 gift card: \( L_2 = 3 \)

Now, let \( n \) be the number of 30-day periods.

A. Write and solve an equation for the number of 30-day periods until the value of the gift cards will be equal.

The value of the $85 gift card after \( n \) periods is given by \( G_1 - n \cdot L_1 \), and the value of the $75 gift card after \( n \) periods is given by \( G_2 - n \cdot L_2 \).

So, the equation for the number of periods until the values are equal is:

\[ G_1 - n \cdot L_1 = G_2 - n \cdot L_2 \]

Substitute in the values:

\[ 85 - 3.50n = 75 - 3n \]

Now, solve for \( n \):

\[ 85 - 3.50n = 75 - 3n \]

\[ 0.50n = 10 \]

\[ n = 20 \]

B. What will the value of each card be when they have equal value?

Substitute \( n = 20 \) into either \( G_1 - n \cdot L_1 \) or \( G_2 - n \cdot L_2 \). Let's use \( G_1 - n \cdot L_1 \):

\[ G_1 - 20 \cdot L_1 \]

\[ 85 - 20 \cdot 3.50 \]

\[ 85 - 70 \]

\[ 15 \]

So, the value of each card will be $15 when they have equal value after 20 periods.