To determine how many cups of coffee Brittany would need to buy for both options to be the same price, let's first lay out the costs for each option.
Option 1: Buying Coffee from Starbucks
- Cost per cup: $6.50
- She has a $50 gift card, so her effective spending is reduced initially.
To find out how many cups she can buy with the gift card:
- Let \( x \) be the number of cups bought.
- The total cost for \( x \) cups would be \( 6.50x \).
- After using the gift card, she only pays the remaining amount after the gift card is exhausted.
The gift card can cover:
\[ \text{Cups covered by gift card} = \frac{50}{6.50} \approx 7.69 \text{ cups} \]
Practically, she can buy 7 cups fully paid by the gift card ($6.50 * 7 = $45.50) and would need to pay for one more cup ($6.50) after that.
Option 2: Buying a Coffeemaker
- Initial cost of the coffeemaker: $200
- Cost per cup at home: $0.55
Let \( y \) be the number of cups she drinks at home.
The cost for \( y \) cups of coffee at home will then be:
\[ \text{Total cost for coffee maker} = 200 + 0.55y \]
Setting the Options Equal
To find when both costs are equal, we set the total costs equal to each other:
\[ 6.50x = 200 + 0.55y \]
Understanding the Math
Let's establish \( y \) and \( x \) in a practical manner: she can buy coffee until her gift card is exhausted and then switch to making coffee at home.
After using her gift card, if she pays out of pocket for the cups beyond the gift card, we can adjust our equation.
Number of Cups
Let's also consider the number of cups after the gift card:
If she drinks 8 cups, she is covered with the gift card for 7, but then the 8th cup requires an out-of-pocket expense:
\[ \text{Cost of 8 cups from Starbucks} = 6.50(8) = 52.00 \] She would exhaust the gift card and pay $6.50 for the 8th cup, leading to a total expense of $52.
Solution
To equate this with the coffeemaker: \[ 6.50x = 200 + 0.55x \]
However, we can simplify to focus on how many cups she would buy before the break-even point: This leads us to isolate \( x \):
\[ 6.50x - 0.55x = 200 \] \[ 5.95x = 200 \] \[ x \approx 33.56 \]
Conclusion
Brittany would need to buy approximately 34 cups of coffee from Starbucks for both options to be the same price.
Choice Recommendation
- If Brittany anticipates drinking more than 34 cups in a reasonable time frame, buying the coffeemaker is likely the better long-term option, considering both the initial investment and the dramatically lower cost per cup afterward ($0.55).
- Conversely, if she thinks she will drink fewer than 34 cups or enjoys the experience at Starbucks enough to justify the price, continuing to buy coffee there might be a better choice for her immediate enjoyment.
Ultimately, the decision should be based on Brittany's coffee consumption habits and how much value she places on the convenience of brewing coffee at home versus the Starbucks experience.