Question

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Part 1
Tavon has a gift card for $ 115 that loses $ 2.50 for each​ 30-day period it is not used. He has another gift card for $ 95 that loses $ 2 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

To solve the problem, we can set up equations for the value of each gift card after a certain number of 30-day periods.

**Part a: Write and solve the equation**

Let \( x \) represent the number of 30-day periods.

For Tavon's first gift card, which starts at $115 and loses $2.50 every 30 days, the value after \( x \) periods can be expressed as:

\[
V_1 = 115 - 2.5x
\]

For the second gift card, starting at $95 and losing $2 every 30 days, the value after \( x \) periods can be expressed as:

\[
V_2 = 95 - 2x
\]

To find when the values of the two gift cards are equal, we set \( V_1 \) equal to \( V_2 \):

\[
115 - 2.5x = 95 - 2x
\]

Now, let's solve this equation:

1. Start by isolating the variable \( x \):
\[
115 - 95 = 2.5x - 2x
\]
\[
20 = 0.5x
\]

2. Now, divide both sides by 0.5:
\[
x = \frac{20}{0.5} = 40
\]

So, the number of 30-day periods until the values of the gift cards are equal is \( x = 40 \).

**Part b: What will the value of each card be when they have equal value?**

Now, we need to substitute \( x = 40 \) back into either equation \( V_1 \) or \( V_2 \) to find the value of the gift cards at that point.

Using \( V_1 \):
\[
V_1 = 115 - 2.5(40)
\]
\[
V_1 = 115 - 100 = 15
\]

Using \( V_2 \) to verify:
\[
V_2 = 95 - 2(40)
\]
\[
V_2 = 95 - 80 = 15
\]

Therefore, when they have equal value, the value of each gift card will be **$15**.

**Summary:**
a. The number of 30-day periods until the value of the gift cards is equal is 40.
b. The value of each card at that time will be $15.