Karen got an annual membership at a gym for 75 dollars to receive a 12 percent discount, after adding a 2.4 percent sales tax, when buying supplements at the gym. Nonmembers receive a 3 percent discount on the cost of supplements after a 2.4 percent sales tax has been added. What is the approximate cost of supplements that Karen must buy annually before adding sales tax for her discount to cover the cost of her membership?
Our first step here is to let 𝑣 be the cost of supplements that Karen buys. There is a 2.4 percent sales tax. 100 percent plus 2.4 percent is equal to 102.4 percent. This is equal to 1.024 as a decimal. The cost of supplements after sales tax but before the discount is 1.24𝑣. We also know that Karen gets 12 percent discount as a member. 100 percent minus 12 percent is equal to 88 percent, which, written as a decimal, is 0.88. The cost of supplements after the discount is 1.24𝑣 multiplied by 0.88.
1.024 multiplied by 0.88 is equal to 0.90112. Therefore, the cost of supplements for Karen after sales tax has been added and the discount has been applied, is 0.90112𝑣. Nonmembers must pay the same sales tax as members. However, they only get a discount of 3 percent. And 100 percent minus 3 percent is equal to 97 percent. This means that the cost of supplements for
members [nonmembers] is 1.024𝑣 multiplied by 0.97. This is equal to 0.99328𝑣.
We know that the annual membership is 75 dollars. We now need to work out what value of 𝑣 will cover this 75-dollar membership. We can calculate the difference in cost between nonmembers and members by subtracting 0.90112𝑣 from 0.99328𝑣. We need to calculate when this is equal to 75 dollars. Subtracting 0.90112 from 0.99328 gives us 0.09216. Therefore, 0.09216𝑣 is equal to 75. Finally, we need to divide both sides by 0.09216. On the left-hand side, they cancel, leaving us with 𝑣. On the right-hand side, we get an answer of 813.802 etcetera.
We were asked to calculate the approximate cost. Therefore, it is sensible to round this to the nearest dollar. As the eight after the decimal point is greater than five, we will round up. Karen needs to buy 814 dollars of supplements for her discount to cover the cost of her membership.
We could check this answer by substituting 814 back into the expression for Karen and for nonmembers. 0.90112 multiplied by 814 is equal to 733.51. 0.99328 multiplied by 814 is equal to 808.53. Rounding these numbers to the nearest dollar gives us 734 and 809 dollars, respectively. Adding 75 on to 734 gives us an answer of 809. This means that the difference in cost is the value of the membership. And our answer of 814 dollars was correct.