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