Problem 22 - Algebra Form and Function

Question

The sales tax on an item is \(6 \%\). Express the total cost, \(C,\) in terms of the price of the item, \(P\).

Step-by-Step Solution

Answer

Answer: C = 1.06 * P

Step 1: Find the sales tax amount

To find the sales tax amount, calculate 6% of the price (P) of the item. This can be expressed as a fraction or decimal: 6% = 0.06. Multiply the price by 0.06 to find the sales tax amount: Sales Tax = 0.06 * P

Step 2: Determine the total cost

To determine the total cost (C) of the item, add the sales tax amount to the price of the item (P): Total Cost (C) = Price (P) + Sales Tax Now, we can substitute the expression for the sales tax that we found in step 1: C = P + (0.06 * P)

Step 3: Simplify the equation

Since both terms in the equation involve the price (P), we can factor it out to simplify the equation: C = P(1 + 0.06) The equation now clearly shows the relationship between the total cost (C) and the price of the item (P). The total cost is equal to the price of the item plus 6% of the price. C = 1.06 * P

Key Concepts

Linear Equations

Understanding linear equations can greatly simplify real-life problems, such as calculating total costs inclusive of sales tax. In this exercise, the final expression for the total cost, \(C\), in terms of the item's price, \(P\), is \(C = 1.06P\). This equation is linear because \(C\) and \(P\) are directly proportional.
Linear equations often look like \(y = mx + b\), where \(m\) is the slope (or rate of change), and \(b\) is the y-intercept. In our case, since there's no fixed additional cost besides the tax, \(b = 0\), and \(m = 1.06\). This demonstrates that for every dollar of price \(P\), the cost \(C\) increases by a factor of 1.06 to account for the 6% sales tax.
Linear equations are extremely useful as they allow us to predict and calculate values easily, by inputting any given price to quickly discover the total cost.

Percentage Calculation

Percentage calculations are essential in determining parts of a whole, which is exactly what sales tax is—a part of the entire price. Here, the sales tax of 6% means that you pay an additional 6% on top of the original price.
To calculate the sales tax itself, we convert the percentage into a decimal form for ease of calculation. The 6% becomes 0.06. Next, you multiply this by the initial price \(P\) to find how much extra you are paying:

• Sales Tax = 0.06 * \(P\)

By understanding how percentages work, you can calculate sales tax, discounts, interest rates, and more which are commonly encountered in daily financial decisions. Remember, percentages are just another way of expressing fractions, where 100% is equivalent to the entire value or quantity.

Price Formulation

Price formulation is crucial for understanding the value exchanges involved in sales transactions. In this exercise, we formulated the final price of an item by incorporating sales tax into the initial price.
The final equation \(C = 1.06P\) shows how the total cost \(C\) is built upon the base price \(P\) with an added percentage from sales tax. It's formulated by first determining the sales tax with

• 0.06 * \(P\)

Then, adding this value to the original item price. This approach of price formulation ensures that no steps are skipped in the price determination process, representing a complete and clear understanding of how additional costs, like tax, accumulate on an initial price.
It's also worth noting that price formulation isn't limited to taxes. Discounts, import fees, and commissions are other elements that might be added or subtracted, following similar principles.