In a world where savvy shopping reigns supreme, Tia found herself drawn to a captivating sweater, its allure amplified by a tempting 20% discount. This savvy shopper recognized the opportunity to save a significant sum while indulging in her fashion desires. How much did she save, and what was the original price of this coveted item? Let's embark on a journey of discovery and unravel the intricacies of Tia's smart purchase.
Tia's keen eye for value propelled her to seize the discounted sweater, securing a remarkable $4.60 savings. This astute move ignited curiosity, prompting us to unravel the original price of the sweater. Join us as we delve into the mathematical calculations that will unveil the initial cost of this fashionable find.
Before we embark on the mathematical expedition, let's establish the fundamental concept of percentage discounts. A percentage discount, often expressed as a fraction or decimal, represents a reduction in the original price. In Tia's case, the 20% discount translates to a 0.2 reduction in price.
Tia bought a sweater that was discounted for 20% of the original price. She saved $4.60 with the discount. What was the original price of the sweater?
To uncover the original price of Tia's discounted sweater, we must navigate the intricacies of percentages and discounts. Here's a step-by-step breakdown of the key points:
- 20% discount
- Saved $4.60
- Original price unknown
- Percentage discount as fraction
- Original price and discount
- Create an equation
- Solve for original price
- Reveal the original price
- Tia's savvy shopping
By understanding these key points and following the mathematical steps, we can unravel the mystery of the original sweater price, shedding light on Tia's successful shopping strategy.
20% discount
The 20% discount that Tia encountered is a crucial piece of information in unraveling the original price of the sweater. A discount, often expressed as a percentage, represents a reduction in the original price, enticing customers with a lower cost.
- Discount as a fraction:
To work with the discount mathematically, we convert it to a fraction. 20% can be expressed as 20/100 or 0.2.
- Discount applied to original price:
The discount is applied to the original price, resulting in a reduced selling price. This relationship can be expressed as: Selling Price = Original Price - Discount.
- Discount amount:
The discount amount is the difference between the original price and the selling price. In Tia's case, this amount is $4.60, which represents the savings she enjoyed.
- Original price and discount:
The original price and the discount are inversely related. A higher discount leads to a lower selling price, and vice versa. This relationship is vital in determining the original price.
By understanding the concept of a discount and its impact on the original price, we can proceed to construct an equation that will unveil the original price of Tia's discounted sweater.
Saved $4.60
Tia's savings of $4.60 is a crucial piece of information that allows us to calculate the original price of the sweater. This saved amount represents the difference between the original price and the discounted price she paid.
To understand the significance of this savings, let's delve into the concept of percentage discounts. When a retailer offers a 20% discount, it means that the customer pays 80% of the original price. This relationship can be expressed mathematically as:
Selling Price = Original Price × (1 - Discount Percentage)
In Tia's case, the discount percentage is 20%, which is equivalent to 0.2. Substituting these values into the equation, we get:
Selling Price = Original Price × (1 - 0.2)
Selling Price = Original Price × 0.8
This equation tells us that the selling price is 80% of the original price. Since Tia saved $4.60, we can infer that the difference between the original price and the selling price is $4.60.
Therefore, we can set up an equation to solve for the original price:
Original Price - Selling Price = $4.60
Original Price - (0.8 × Original Price) = $4.60
Simplifying this equation, we get:
0.2 × Original Price = $4.60
Dividing both sides by 0.2, we finally arrive at the original price:
Original Price = $4.60 ÷ 0.2
Original Price = $23
Hence, the original price of the sweater before the discount was $23.
By carefully analyzing the information about Tia's savings and the discount percentage, we were able to unravel the original price of the sweater, demonstrating the power of mathematical reasoning in solving real-world problems.
Original price unknown
The original price of the sweater is initially unknown, represented by the variable "x". Our goal is to solve for this unknown variable using the information provided in the problem.
- Discount as a fraction:
The 20% discount can be expressed as a fraction or decimal. As a fraction, it is 20/100 or 0.2. This fraction represents the proportion of the original price that is discounted.
- Discount applied to original price:
The discount is applied to the original price, resulting in a reduced selling price. This relationship can be expressed mathematically as:
Selling Price = Original Price - Discount
Selling Price = x - (0.2 * x)
where "x" represents the original price.
- Discount amount:
The discount amount is the difference between the original price and the selling price. In this case, the discount amount is $4.60.
- Original price and discount:
The original price and the discount are inversely related. A higher discount leads to a lower selling price, and vice versa. This relationship is crucial in determining the original price.
By understanding these concepts and setting up an equation, we can solve for the original price, which is the ultimate goal of this problem.
Percentage discount as fraction
Expressing a percentage discount as a fraction is a crucial step in solving this problem mathematically. It allows us to manipulate the discount in calculations more easily.
- Definition of fraction:
A fraction represents a part of a whole. It consists of two numbers: the numerator and the denominator. The numerator is the number above the line, and the denominator is the number below the line.
- Converting percentage to fraction:
To convert a percentage to a fraction, we divide the percentage by 100. For example, to convert 20% to a fraction, we divide 20 by 100, which gives us 20/100.
- Simplifying fraction:
We can often simplify fractions by dividing both the numerator and the denominator by a common factor. For example, the fraction 20/100 can be simplified by dividing both numbers by 20, which gives us 1/5.
- Fraction representing discount:
In this problem, the 20% discount can be expressed as the fraction 1/5. This means that the discount is equal to one-fifth of the original price.
By converting the percentage discount to a fraction, we can now use it in mathematical operations to solve for the original price.
Original price and discount
Understanding the relationship between the original price and the discount is fundamental to solving this problem. These two values are inversely related, meaning that as the discount increases, the original price decreases, and vice versa.
- Discount as a reduction in price:
A discount represents a reduction in the original price. When a discount is applied, the customer pays less than the original price.
- Calculating discounted price:
To calculate the discounted price, we subtract the discount amount from the original price. Mathematically, this can be expressed as:
Discounted Price = Original Price - Discount
where "Discount" is the discount amount and "Original Price" is the original price.
- Discount as a percentage of original price:
In this problem, the discount is given as a percentage of the original price. This means that the discount amount is equal to a certain percentage of the original price.
- Solving for original price:
Our goal is to find the original price. To do this, we need to isolate the original price on one side of the equation. We can achieve this by adding the discount amount to both sides of the equation:
Original Price = Discounted Price + Discount
where "Discounted Price" is the selling price after the discount and "Discount" is the discount amount.
By understanding the relationship between the original price and the discount, and using the mathematical equation above, we can solve for the original price.
Create an equation
To solve this problem, we need to set up an equation that represents the relationship between the original price, the discount, and the discounted price. We can start by introducing the following variables:
- x: Original price of the sweater
- y: Discount amount
- z: Discounted price
From the information given in the problem, we know that:
- The discount is 20% of the original price, which can be expressed as y = 0.2x.
- Tia saved $4.60 with the discount, which means the discounted price is $4.60 less than the original price: z = x - y.
We can use these two equations to create a single equation that will allow us to solve for x, the original price of the sweater.
Substituting the first equation into the second equation, we get:
z = x - (0.2x)
z = 0.8x
Now we have an equation that relates the discounted price z to the original price x. We can use this equation to find the value of x.
In the next step, we will rearrange the equation to solve for x and find the original price of the sweater.
Solve for original price
To solve for the original price x, we need to rearrange the equation z = 0.8x that we derived in the previous step.
To isolate x on one side of the equation, we can divide both sides by 0.8. This gives us:
x = z / 0.8
Now we can substitute the value of z, which is the discounted price, into the equation.
In this problem, Tia saved $4.60 with the discount, so the discounted price z is $4.60 less than the original price x. Therefore, we can write:
z = x - 4.60
Substituting this into the equation above, we get:
x = (x - 4.60) / 0.8
Now we can solve for x by multiplying both sides of the equation by 0.8.
0.8x = x - 4.60
0.2x = -4.60
x = -4.60 / 0.2
x = 23
Therefore, the original price of the sweater was $23.
Reveal the original price
To reveal the original price of the sweater, we need to solve the equation x = -4.60 / 0.2 that we derived in the previous step.
- Isolating x:
Our goal is to isolate x, which represents the original price, on one side of the equation. To do this, we can multiply both sides of the equation by 0.2.
- Solving for x:
Multiplying both sides of the equation by 0.2 gives us: 0.2x = -4.60. Dividing both sides by 0.2, we get: x = -4.60 / 0.2.
- Calculating the value of x:
Evaluating the expression -4.60 / 0.2, we find that x = 23.
- Original price revealed:
Therefore, the original price of the sweater before the discount was $23.
Tia's clever shopping strategy allowed her to save $4.60 on the original price of the sweater, which was $23.
Tia's savvy shopping
Tia's shopping strategy exemplifies the art of making informed and value-conscious purchases.
- Recognizing the discount opportunity:
Tia's keen eye for value prompted her to identify the 20% discount as an opportunity to save money on her desired sweater.
- Calculating the savings:
Before making the purchase, Tia calculated the amount she would save with the discount. This foresight ensured that she understood the true value of her purchase.
- Making a smart decision:
Armed with the knowledge of the savings, Tia made a well-informed decision to purchase the sweater, confident that she was getting a good deal.
- Enjoying the benefits of her purchase:
Not only did Tia save money, but she also acquired a stylish sweater that she genuinely liked. Her savvy shopping allowed her to indulge in her fashion desires while being mindful of her budget.
Tia's shopping experience serves as a reminder that with a bit of planning and attentiveness, consumers can make purchases that align with their financial goals and personal preferences.
FAQ
Introduction:
Here are some frequently asked questions and answers related to the scenario of Tia's discounted sweater purchase:
Question 1: What was the original price of the sweater before the discount?
Answer 1: The original price of the sweater was $23.
Question 2: How much did Tia save with the 20% discount?
Answer 2: Tia saved $4.60 with the 20% discount.
Question 3: How was the original price calculated?
Answer 3: The original price was calculated by dividing the discounted price by the discount rate. In this case, we divided the discounted price of $18.40 by the discount rate of 0.2 (20% expressed as a decimal).
Question 4: Can you explain the concept of percentage discounts?
Answer 4: A percentage discount is a reduction in the original price of an item, typically expressed as a percentage. For example, a 20% discount means that the item is being sold for 80% of its original price.
Question 5: Are there any other ways to calculate the original price?
Answer 5: Yes, there are a few other ways to calculate the original price. One method is to use the following formula: Original Price = Discounted Price / (1 - Discount Rate). Another method is to use the proportion Original Price : Discounted Price = 1 : (1 - Discount Rate).
Question 6: How can I apply this knowledge to my own shopping experiences?
Answer 6: By understanding percentage discounts and knowing how to calculate the original price, you can make informed decisions when shopping. You can compare prices and discounts from different stores to get the best deal on the items you want.
Closing Paragraph:
We hope these questions and answers have provided you with a clearer understanding of the concepts related to Tia's discounted sweater purchase. If you have any further questions, feel free to ask.
Now that we have covered the frequently asked questions, let's explore some additional tips that can help you make smart and savvy shopping decisions.
Tips
Introduction:
Here are some practical tips to help you make the most of discounts and shop wisely:
Tip 1: Do your research:
Before making a purchase, take some time to research the item you want to buy. Check different stores and online retailers to compare prices and see if there are any discounts or promotions available. This will help you find the best deal.
Tip 2: Understand the different types of discounts:
There are various types of discounts, such as percentage discounts, flat discounts, and buy-one-get-one-free offers. Make sure you understand how each type of discount works so that you can take advantage of the best deals.
Tip 3: Calculate the actual savings:
Don't just focus on the discount percentage. Calculate the actual amount of money you will save by purchasing the item at a discounted price. This will help you determine if the discount is truly worth it.
Tip 4: Consider the quality of the item:
While discounts can be tempting, don't compromise on the quality of the item you are buying. Make sure you inspect the item carefully and read reviews from other customers before making a purchase.
Closing Paragraph:
By following these tips, you can become a savvy shopper who makes informed decisions and gets the best value for your money.
In conclusion, understanding percentage discounts and applying these practical tips will empower you to make smart shopping choices, just like Tia did when she purchased her discounted sweater.
Conclusion
In the realm of savvy shopping, Tia's purchase of a discounted sweater exemplifies the art of making informed and value-conscious decisions.
We embarked on a mathematical journey, unraveling the intricacies of percentage discounts and their impact on the original price. Through a series of logical steps, we calculated the original price of the sweater, revealing that Tia saved a significant $4.60 thanks to the 20% discount.
This exercise not only provided a solution to the problem but also highlighted the importance of understanding the concepts underlying discounts and pricing strategies. As consumers, we can empower ourselves by asking questions, comparing prices, and making calculated choices that align with our financial goals.
Just like Tia, we can all become discerning shoppers, seeking out bargains and making purchases that bring us both satisfaction and savings. Remember, a little bit of research and a keen eye for value can go a long way in helping us make smart and informed shopping decisions.
Closing Message:
May we all have the wisdom of Tia, the next time we encounter a captivating discounted item, ensuring that we make purchases that truly reflect our needs and desires, while also staying within our budget.
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