Which Two Expressions Correctly Calculate The Total Amount Camacho Pays For The Truck, Including A 13% Tax?

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#h1 Tax and Tip Word Problems: Mastering Total Cost Calculations

When dealing with real-world financial scenarios, understanding how to calculate the total cost, including taxes and tips, is essential. This article delves into the intricacies of such calculations, providing a comprehensive guide to solving tax and tip word problems. We'll explore various expressions and methods to accurately determine the final amount you'll pay for goods or services. Let's consider a practical example to illustrate these concepts.

Understanding the Basics of Tax and Tip Calculations

When you purchase an item, the final cost often includes not only the original price but also additional charges such as sales tax and, in some cases, a tip. Sales tax is a percentage of the item's price that is added to the total cost and is collected by the government. A tip, on the other hand, is a voluntary payment added to the bill for services rendered, typically in restaurants or for other service-based industries. Calculating these additions accurately is crucial for budgeting and financial planning.

Key Concepts in Tax and Tip Calculations

  1. Original Price (x): This is the initial cost of the item or service before any additional charges.
  2. Tax Rate (as a decimal): The sales tax rate is usually given as a percentage, but to perform calculations, it must be converted to a decimal. For example, a 13% tax rate is equivalent to 0.13.
  3. Tip Rate (as a decimal): Similar to the tax rate, the tip rate is also expressed as a percentage and needs to be converted to a decimal for calculations. For instance, a 15% tip rate is 0.15.
  4. Tax Amount: This is the amount of tax added to the original price, calculated by multiplying the original price by the tax rate.
  5. Tip Amount: The tip amount is calculated by multiplying the original price (or sometimes the price including tax) by the tip rate.
  6. Total Cost: The total cost is the sum of the original price, the tax amount, and the tip amount.

Formulas for Calculation

To better understand these concepts, let's outline the formulas involved:

  • Tax Amount = Original Price × Tax Rate
  • Tip Amount = Original Price × Tip Rate
  • Total Cost = Original Price + Tax Amount + Tip Amount

Applying the Concepts: Camacho's Truck Purchase

Consider a scenario where Camacho is purchasing a truck, and we need to determine the total cost, including sales tax. The problem presents several expressions and asks us to identify which ones accurately represent the total amount Camacho pays.

Problem: Which of the following expressions could represent how much Camacho pays in total for the truck?

Choose 2 answers:

A.

(1+0.13) x

B.

13/100 x

C.

13 x

D.

1.  13 x

E.

13 x + x

Breaking Down the Problem

To solve this problem, we need to understand how each expression relates to the original price of the truck and the sales tax. Let's assume the original price of the truck is represented by x. The sales tax is 13%, which means we need to add 13% of the original price to the original price to find the total cost.

Analyzing the Options

  • Option A: (1 + 0.13)x
    • This expression represents the total cost by adding 1 (representing 100% of the original price) to 0.13 (representing 13% sales tax) and multiplying the sum by the original price (x). This is a valid way to calculate the total cost.
  • Option B: (13/100)x
    • This expression calculates only the tax amount (13% of x) but does not include the original price. Therefore, it does not represent the total cost.
  • Option C: 13x
    • This expression multiplies the original price by 13, which does not logically represent adding a 13% tax. It's an incorrect representation of the total cost.
  • Option D: 1.13x
    • This expression is equivalent to option A. It multiplies the original price (x) by 1.13, which represents the original price plus 13% tax. This is also a valid way to calculate the total cost.
  • Option E: 13x + x
    • This expression is incorrect because it multiplies the original price by 13 and then adds the original price. This does not accurately represent a 13% tax calculation.

Correct Answers

The correct answers are A. (1 + 0.13)x and D. 1.13x. Both expressions accurately represent the total cost of the truck, including the 13% sales tax.

Deeper Dive into Percentage Calculations

Understanding percentages is fundamental to solving tax and tip problems. A percentage is a way of expressing a number as a fraction of 100. For instance, 13% means 13 out of 100. When calculating percentages, it’s essential to convert the percentage to its decimal form by dividing it by 100. This conversion is critical for accurate calculations.

Converting Percentages to Decimals

To convert a percentage to a decimal, divide the percentage by 100. For example:

  • 13% = 13 / 100 = 0.13
  • 7% = 7 / 100 = 0.07
  • 15% = 15 / 100 = 0.15

This conversion allows us to easily multiply the original price by the decimal to find the tax or tip amount. Conversely, to convert a decimal back to a percentage, you multiply the decimal by 100.

Calculating Tax and Tip Amounts

Once the percentage is converted to a decimal, calculating the tax or tip amount is straightforward. You multiply the original price by the decimal equivalent of the tax or tip rate.

Example:

If an item costs $50 and the sales tax is 8%, the tax amount is calculated as follows:

  • Tax Rate (decimal) = 8 / 100 = 0.08
  • Tax Amount = $50 × 0.08 = $4

Similarly, if you want to leave a 20% tip on a $50 meal:

  • Tip Rate (decimal) = 20 / 100 = 0.20
  • Tip Amount = $50 × 0.20 = $10

Combining Tax and Tip Calculations

In some scenarios, you may need to calculate both tax and tip. The standard approach is to calculate the tax first and add it to the original price. Then, the tip is calculated on the sum of the original price and the tax. This ensures that the tip is based on the total cost of the service, including tax.

Example:

Suppose you have a restaurant bill of $80, the sales tax is 7%, and you want to leave a 18% tip. Here’s how you would calculate the total cost:

  1. Calculate the tax amount:
    • Tax Rate (decimal) = 7 / 100 = 0.07
    • Tax Amount = $80 × 0.07 = $5.60
  2. Add the tax to the original price:
    • Price with Tax = $80 + $5.60 = $85.60
  3. Calculate the tip amount:
    • Tip Rate (decimal) = 18 / 100 = 0.18
    • Tip Amount = $85.60 × 0.18 = $15.41 (rounded to the nearest cent)
  4. Calculate the total cost:
    • Total Cost = $85.60 + $15.41 = $101.01

Advanced Scenarios and Complex Calculations

While the basic calculations for tax and tips are straightforward, some scenarios may involve more complex considerations. For example, some regions have tiered tax systems where different items are taxed at different rates. Additionally, some service industries may have suggested tip amounts based on the level of service provided.

Handling Tiered Tax Systems

In a tiered tax system, different categories of goods or services may be subject to different tax rates. For instance, groceries might be taxed at a lower rate than luxury goods. When dealing with such systems, it’s essential to calculate the tax for each category separately and then add them together to find the total tax amount.

Example:

Suppose you purchase groceries worth $100, taxed at 3%, and clothing worth $50, taxed at 8%. The calculations would be:

  1. Groceries Tax:
    • Tax Amount = $100 × (3 / 100) = $3
  2. Clothing Tax:
    • Tax Amount = $50 × (8 / 100) = $4
  3. Total Tax = $3 + $4 = $7

Tips on Services

In service industries, tip amounts can vary based on the quality of service. While a standard tip is often around 15-20%, exceptional service may warrant a higher tip, and unsatisfactory service may result in a lower tip or none at all. When solving word problems involving tips, it’s important to carefully consider the context and any specific instructions provided.

Using Algebraic Expressions

Algebraic expressions are a powerful tool for representing and solving tax and tip problems. As demonstrated in the initial problem with Camacho's truck purchase, understanding how to form and interpret algebraic expressions can simplify complex calculations.

Consider a general scenario where you want to calculate the total cost of an item with a tax rate of t (as a decimal) and a tip rate of p (as a decimal). If the original price is x, the total cost can be expressed as:

Total Cost = x + tx + p( x + tx )

This expression can be simplified as:

Total Cost = x(1 + t) + p x(1 + t)

Total Cost = x(1 + t)(1 + p)

This general formula can be applied to various scenarios by substituting the appropriate values for x, t, and p.

Practice Problems and Solutions

To solidify your understanding, let’s work through a few practice problems.

Problem 1:

John buys a laptop for $1200. The sales tax is 6%. What is the total cost of the laptop?

Solution:

  1. Calculate the tax amount:
    • Tax Rate (decimal) = 6 / 100 = 0.06
    • Tax Amount = $1200 × 0.06 = $72
  2. Calculate the total cost:
    • Total Cost = $1200 + $72 = $1272

The total cost of the laptop is $1272.

Problem 2:

Sarah has a dinner bill of $65. She wants to leave a 20% tip. How much will she pay in total?

Solution:

  1. Calculate the tip amount:
    • Tip Rate (decimal) = 20 / 100 = 0.20
    • Tip Amount = $65 × 0.20 = $13
  2. Calculate the total cost:
    • Total Cost = $65 + $13 = $78

Sarah will pay $78 in total.

Problem 3:

A restaurant bill is $150, and the sales tax is 8%. If you want to leave an 18% tip, what is the total amount you will pay?

Solution:

  1. Calculate the tax amount:
    • Tax Rate (decimal) = 8 / 100 = 0.08
    • Tax Amount = $150 × 0.08 = $12
  2. Add the tax to the original price:
    • Price with Tax = $150 + $12 = $162
  3. Calculate the tip amount:
    • Tip Rate (decimal) = 18 / 100 = 0.18
    • Tip Amount = $162 × 0.18 = $29.16
  4. Calculate the total cost:
    • Total Cost = $162 + $29.16 = $191.16

The total amount you will pay is $191.16.

Conclusion

Mastering tax and tip word problems involves understanding the basic concepts of percentages, converting percentages to decimals, and applying the correct formulas. By breaking down problems into smaller steps and using algebraic expressions, you can confidently solve even the most complex scenarios. Remember, practice is key to improving your skills, so work through plenty of examples and real-world applications to become proficient in these essential calculations. Whether you're calculating the final cost of a purchase or determining the appropriate tip for a service, these skills are invaluable in everyday financial situations.

By understanding these calculations, you can make informed decisions and manage your finances effectively. This article has provided a comprehensive guide to tackling tax and tip problems, ensuring you are well-equipped to handle these calculations in any context.