At the wintergrove amusement park you can purchase a season discount pass for 75$ that will allow a 10$ admission to the park all season. If you do not purchase a season discount pass, then the park admission is 35$. Determine the number of times you must visit the park with a pass that would equal the cost of paying full price.

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To determine the number of times you must visit the park with a season discount pass in order to equal the cost of paying full price, we need to compare the total cost of using the pass versus the total cost of paying full price each time.

First, let's calculate the cost of visiting the park with a season discount pass.

The cost of the pass itself is $75.

For each visit to the park, the discounted admission price is $10.

Let's assume the number of visits with the season discount pass is represented by n.

The total cost of visiting the park with the season discount pass would be:

Total Cost with Pass = Cost of Pass + (Number of Visits with Pass * Discounted Admission Price)
= $75 + ($10*n)
= $75 + $10n

Now, let's calculate the cost of paying full price each time.

The admission price without the season discount pass is $35.

Let's assume the number of visits without the pass is represented by m.

The total cost of visiting the park without the season discount pass would be:

Total Cost without Pass = Number of Visits without Pass * Full Price Admission
= $35 * m
= $35m

To find the number of visits that would equal the cost of paying full price, we need to set the two total costs equal to each other:

$75 + $10n = $35m

To simplify the equation, we can divide both sides by 5:

$15 + $2n = $7m

Now, we can analyze the equation.

If we consider whole numbers for n and m, we can start by finding the smallest value for n that makes the equation true. We can then solve for m.

Let's begin by testing n = 1:

$15 + $2(1) = $7m
$15 + $2 = $7m
$17 = $7m

Since 17 is not divisible evenly by 7, n = 1 does not give a whole number value for m.

Let's try n = 2:

$15 + $2(2) = $7m
$15 + $4 = $7m
$19 = $7m

Again, 19 is not divisible evenly by 7, so n = 2 does not give a whole number value for m.

We can keep trying higher values of n until we find a whole number value for m. Alternatively, we can rearrange the equation to solve for m:

$2n - $7m = $15

Using trial and error, we can eventually find a pair of values for n and m that makes the equation true. In this case, there is no whole number solution. Thus, there is no specific number of visits that would equal the cost of paying full price.