Question

Business: Discount Pricing A car dealer, at a year-end clearance, reduces the list price of last year's models by \(15 \%\). If a certain four-door model has a discounted price of 18,000 dollar, what was its list price? How much can be saved by purchasing last year's model?

Short answer

The list price was approximately 21,176.47 dollars. The savings by purchasing last year's model is approximately 3,176.47 dollars.

Step-by-step solution

Define the Known Values

Identify the known values provided in the problem. The discount percentage is given as 15% and the discounted price of the car is 18,000 dollars.

Represent the Discount Relationship

Let the list price of the car be represented by the variable P. The discounted price can be expressed using the formula for discount: \[ P_{\text{discounted}} = P - 0.15P = 0.85P \]Given that the discounted price is 18,000 dollars, set up the equation: \[ 0.85P = 18000 \]

Solve for the List Price

To find the list price (P), solve the equation for P by dividing both sides by 0.85: \[ P = \frac{18000}{0.85} \]Calculate the value: \[ P \thickapprox 21176.47 \]

Calculate the Savings

Find out how much is saved by purchasing last year's model. Subtract the discounted price from the list price: \[ \text{Savings} = P - \text{Discounted Price} = 21176.47 - 18000 \] Calculate the value: \[ \text{Savings} \thickapprox 3176.47 \]

Key concepts

percentage discount

Let's start by understanding what a percentage discount is. When you buy something at a discount, you're paying less than the original price. The reduction is represented as a percentage of the original price. In this problem, the car dealer offers a 15% discount on last year's models, meaning the buyer pays 15% less than the original price.

A percentage discount can be calculated with the formula:
\[ \text{Discount Amount} = \text{Original Price} \times \frac{\text{Discount Percentage}}{100} \]
So, if we are given a discount percentage and a discounted price, we can use these values to figure out the list price, or original price.

list price calculation

The next step is understanding how to calculate the list price, which is the original price before any discounts. In this exercise, we let the list price be represented by the variable P.

Given the discount relationship where the discounted price is 85% of the list price (because 100% - 15% = 85%), we can write the formula:
\[ P_{\text{discounted}} = P - 0.15P = 0.85P \]
Now, if we know the discounted price is 18,000 dollars, we set up the following equation to find the list price:
\[ 0.85P = 18000 \]
Solving for P involves dividing both sides by 0.85. This gives us:
\[ P = \frac{18000}{0.85} \]
Simplifying that, we find the list price to be approximately 21,176.47 dollars.

savings calculation

Now, let’s calculate how much you save by buying the car at the discounted price. Your savings is the difference between the list price and the discounted price.

We already found the list price to be approximately 21,176.47 dollars. The savings can be calculated as:
\[ \text{Savings} = P - \text{Discounted Price} = 21176.47 - 18000 \]
Subtracting these amounts, we get:
\[ \text{Savings} \thickapprox 3176.47 \]
This means you save approximately 3,176.47 dollars by purchasing the car at the discounted price.

algebraic equations in pricing

Algebraic equations are very useful when dealing with pricing problems, especially in discounts and savings. They help you systematically find the unknown values based on the information provided.

In this exercise, we used an algebraic equation to express the relationship between the list price and the discounted price. Setting up the equation:
\[ 0.85P = 18000 \]
allowed us to solve for the list price (P) by performing algebraic operations.

We can generalize this method to any discount problem:

• Represent the list price with a variable (e.g., P).
• Use the discount percentage to form an equation.
• Solve the equation for the list price or any other unknown value.

This process makes it straightforward to handle more complex pricing scenarios with ease.