A local gym offers two choices in group discounts. Package A has a one-time enrollment fee of $205.59 per person and a monthly fee of $39.63. There is also an optional $15.48 monthly fee for full access to the swimming pool. Package B has a one-time enrollment fee of $185.64 per person and a monthly fee of $59.25. There is also an optional $22.38 monthly fee for full access to the swimming pool. Which package is best for a couple who wants to join the gym and utilize the pool for five years?

Step 1: Assign variable for the unknown

Let us try to identify the cost involved in each package. Let x be the number of people and y be the number of months.

Step 2: Identify expression for the given information

We need to write an expression to denote the total cost involved to join the gym and utilize the swimming pool for ' x ' number of people for 'y' number of months.

Statement 1: Package A has a one-time enrollment fee of $205.59 per person and a monthly fee of $39.63. There is also an optional $15.48 monthly fee for full access to the swimming pool.

So based on the statement 1

Cost for Package A = [205.59 + (39.63+15.48) y]x

Statement 2: Package B has a one-time enrollment fee of $185.64 per person and a monthly fee of $59.25. There is also an optional $22.38 monthly fee for full access to the swimming pool.

So based on the statement 2

Cost for Package B = [185.64 + (59.25+22.38) y]x

Step 3: Simplifying each expression

Package A = [205.59 + (39.63+15.48) y]x = [205.59+55.11y]x

Package B = [185.64 + (59.25+22.38) y]x = [185.64+81.63y]x

Step 4: Evaluate each expression

For a couple, Number of people (x) = 2, and for 5 years, Number of months (y) = 5 years * (12 month/year) = 60

We need to substitute 2 for x and 60 for y in the expression and see which package cost is lower to give the correct recommendation.

Package A: [205.59+55.11y]x = [205.59+55.11 (60) ]*2=$7024.38

Package B: [185.64+81.63y]x = [185.64+81.63 (60) ]*2=$10166.88

Conclusion:

Package A is cheaper than Package B. So we can conclude Package A is best for a couple who wants to join the gym and utilize the pool for five year.

A couple who want to join the gym and utilize the pool for five years, would have to pay one-time enrollment fee for two persons, monthly fee of gym and pool for 60 months.

Therefore Total Expenditure = (2 * one-time enrollment fee) + (2 * 60 * monthly fee of gym) + (2 * 60 * monthly fee of pool)

Now, Package A has a one-time enrollment fee of $205.59 per person and a monthly fee of $39.63. There is also an optional $15.48 monthly fee for full access to the swimming pool.

So Expenditure A = (2*205.59) + (2*60*39.63) + (2*60*15.48) = $7024.38

And, Package B has a one-time enrollment fee of $185.64 per person and a monthly fee of $59.25. There is also an optional $22.38 monthly fee for full access to the swimming pool.

So Expenditure B = (2*185.64) + (2*60*59.25) + (2*60*22.38) = $10,166.88

It is clearly visible that Expenditure B is more than Expenditure A. Hence, Package A is best option for the couple.