Solving for the Number of Sweaters

Oct 13, 2025 by miodeya.com 44 views

Introduction

In this article, we will delve into a real-world scenario involving a gift card and explore the mathematical concepts that govern the situation. Favian, the protagonist, has received a $\$100$ gift card to a clothing store and uses it to purchase $n$ sweaters and 1 belt. We will analyze the given information, identify the unknowns, and use mathematical techniques to solve for the number of sweaters Favian purchased.

The Problem

Favian received a $\$100$ gift card to a clothing store. Using only the gift card, he was able to purchase $n$ sweaters that cost $\$32$ each and 1 belt for $\$20$ . If there was no tax on the sweaters and belt, which of the following equations represents the total cost of the items purchased?

Equation 1: Total Cost = Number of Sweaters × Cost per Sweater + Cost of Belt

\text{Total Cost} = 32n + 20

Equation 2: Total Cost = Gift Card Value - Remaining Balance

\text{Total Cost} = 100 - \text{Remaining Balance}

Equation 3: Total Cost = Number of Sweaters × Cost per Sweater + Number of Sweaters × Cost per Sweater + Cost of Belt

\text{Total Cost} = 32n + 32n + 20

Equation 4: Total Cost = Gift Card Value - (Number of Sweaters × Cost per Sweater + Cost of Belt)

\text{Total Cost} = 100 - (32n + 20)

Analyzing the Equations

Let's analyze each equation to determine which one represents the total cost of the items purchased.

Equation 1: Total Cost = Number of Sweaters × Cost per Sweater + Cost of Belt

This equation represents the total cost of the items purchased, as it takes into account the cost of each sweater and the cost of the belt.

Equation 2: Total Cost = Gift Card Value - Remaining Balance

This equation represents the total cost of the items purchased, as it takes into account the gift card value and the remaining balance.

Equation 3: Total Cost = Number of Sweaters × Cost per Sweater + Number of Sweaters × Cost per Sweater + Cost of Belt

This equation is incorrect, as it represents the total cost of the items purchased twice.

Equation 4: Total Cost = Gift Card Value - (Number of Sweaters × Cost per Sweater + Cost of Belt)

This equation is also incorrect, as it represents the total cost of the items purchased in a different way.

Solving for the Number of Sweaters

Now that we have identified the correct equation, let's solve for the number of sweaters Favian purchased.

Step 1: Write the Equation

\text{Total Cost} = 32n + 20

Step 2: Set Up the Equation

32n + 20 = 100

Step 3: Subtract 20 from Both Sides

32n = 80

Step 4: Divide Both Sides by 32

n = \frac{80}{32}

Step 5: Simplify the Fraction

n = \frac{5}{2}

Step 6: Convert the Fraction to a Decimal

n = 2.5

Conclusion

In this article, we explored a real-world scenario involving a gift card and used mathematical techniques to solve for the number of sweaters Favian purchased. We analyzed the given information, identified the unknowns, and used algebraic manipulations to solve for the number of sweaters. The final answer is $n = 2.5$ , which represents the number of sweaters Favian purchased.

Real-World Applications

This problem has real-world applications in various fields, such as finance, economics, and business. For example, a company may use a gift card to purchase items for a marketing campaign, and the company's financial team may need to calculate the total cost of the items purchased. In this scenario, the company's financial team would use the same mathematical techniques to solve for the number of items purchased.

Future Research Directions

Future research directions in this area may include:

Exploring different scenarios: Researchers may explore different scenarios, such as a gift card with a variable value or a gift card with a limited expiration date.
Developing new mathematical models: Researchers may develop new mathematical models to represent the total cost of the items purchased.
Analyzing the impact of taxes: Researchers may analyze the impact of taxes on the total cost of the items purchased.

Introduction

In our previous article, we explored a real-world scenario involving a gift card and used mathematical techniques to solve for the number of sweaters Favian purchased. In this article, we will answer some of the most frequently asked questions related to the problem.

Q&A

Q: What is the total cost of the items purchased?

A: The total cost of the items purchased is $\$100$ , which is the value of the gift card.

Q: How many sweaters did Favian purchase?

A: Favian purchased $n = 2.5$ sweaters, but since you can't buy a fraction of a sweater, we can round down to $n = 2$ sweaters.

Q: What is the cost of each sweater?

A: The cost of each sweater is $\$32$ .

Q: What is the cost of the belt?

A: The cost of the belt is $\$20$ .

Q: What is the remaining balance on the gift card?

A: The remaining balance on the gift card is $\$0$ , since Favian used the entire gift card to purchase the sweaters and belt.

Q: Can Favian purchase more items with the gift card?

A: No, Favian cannot purchase more items with the gift card, since the gift card has been fully used.

Q: What if Favian wants to purchase more sweaters?

A: If Favian wants to purchase more sweaters, he will need to use a different payment method, such as a credit card or cash.

Q: Can Favian use the gift card to purchase items from a different store?

A: No, Favian can only use the gift card to purchase items from the clothing store where the gift card was issued.

Q: What if Favian wants to return one of the sweaters?

A: If Favian wants to return one of the sweaters, he will need to follow the store's return policy and may receive a refund or exchange the sweater for a different item.

Conclusion

In this article, we answered some of the most frequently asked questions related to the problem. We hope that this Q&A article has provided you with a better understanding of the mathematical concepts that govern real-world scenarios and has helped you to apply mathematical techniques to solve problems.

Real-World Applications

Future Research Directions

Future research directions in this area may include:

Exploring different scenarios: Researchers may explore different scenarios, such as a gift card with a variable value or a gift card with a limited expiration date.
Developing new mathematical models: Researchers may develop new mathematical models to represent the total cost of the items purchased.
Analyzing the impact of taxes: Researchers may analyze the impact of taxes on the total cost of the items purchased.

By exploring these research directions, we can gain a deeper understanding of the mathematical concepts that govern real-world scenarios and develop new mathematical models to represent complex situations.

Glossary

Gift card: A card that contains a certain amount of money that can be used to purchase items from a specific store.
Sweater: A type of clothing that is worn to keep warm.
Belt: A type of clothing that is worn around the waist.
Total cost: The total amount of money spent on items purchased.
Remaining balance: The amount of money left on a gift card after it has been used to purchase items.

References

[1] Favian's Gift Card Shopping Spree: A Mathematical Exploration. (2023). Retrieved from https://example.com/favian-gift-card-shopping-spree
[2] Gift Card Industry Report. (2022). Retrieved from https://example.com/gift-card-industry-report

Note: The references provided are fictional and for demonstration purposes only.