Question

Prep Exercise 4 Identify the percent, initial amount, and amount of increase or decrease. A \(25 \%\) discount is applied to the price of an appliance, resulting in a discount of \(\$ 80\). What was the original price of the appliance? Percent:_____ Initial amount:_____ Amount of increase or decrease:_____ A lamp is on sale for \(15 \%\) off. If the discount amount is \(\$ 13.49,\) what is the initial price?

Short answer

Answer: The initial amount for the appliance is $320, and the amount of decrease (discount) is $80. The initial amount for the lamp is $89.93, and the amount of decrease (discount) is $13.49.

Step-by-step solution

Identify the given information

We are given a \(25\%\) discount on the price of an appliance, resulting in a discount of \(\$80\). We need to find the original price of the appliance.

Set up the equation

Use the formula: Discount Amount = Original Price × Discount Percent, which can be represented as \(80 = P \times 0.25\).

Solve for the original price

Divide both sides of the equation by \(0.25\) to solve for the original price (P): \(P = 80 / 0.25\).

Calculate the original price

After dividing, we get \(P = 320\). So, the original price of the appliance was \(\$320\). #Problem 2: Lamp Discount#

Identify the given information

We are given a \(15\%\) discount on a lamp, which results in a discount amount of \(\$13.49\). We need to find the initial price of the lamp.

Set up the equation

Use the formula: Discount Amount = Original Price × Discount Percent, which can be represented as \(13.49 = L \times 0.15\).

Solve for the initial price

Divide both sides of the equation by \(0.15\) to solve for the initial price (L): \(L = 13.49 / 0.15\).

Calculate the initial price

After dividing, we get \(L = 89.93\). So, the initial price of the lamp was \(\$89.93\). In conclusion, for the appliance, the percent discount is \(25 \%\), the initial amount is \(\$320\), and the amount of decrease (discount) is \(\$80\). For the lamp, the percent discount is \(15 \%\), the initial amount is \(\$89.93\), and the amount of decrease (discount) is \(\$13.49\).

Key concepts

Discount Calculation

Calculating a discount is important in understanding how much we save on an item. When a store offers a discount, they reduce the item's original price by a certain percentage. To find the discount, you multiply the original price by the percentage of the discount (in decimal form). For example, if you have a laptop priced at \(200 with a 10% discount, you calculate the discount amount by multiplying \)200 by 0.10, which equals $20.

• Discount Formula: \( \text{Discount Amount} = \text{Original Price} \times \text{Discount Percent} \)
• Convert percentage to decimal: Move the decimal two places left.
• Subtract the discount from the original price for the sale price.

By following these steps, you'll understand your savings and the remaining purchase amount.

Original Price Determination

Determining the original price from a discount involves reversing the discount calculation. When you know the discount amount and the discount percentage, you can find out what the item originally cost before the discount was applied. Let's say you bought an item for \(40 less because there was a 20% discount.

• Start by setting up the equation: \( \text{Discount Amount} = \text{Original Price} \times \text{Discount Percent} \)
• Solve for the original price: Divide the discount amount by the discount percentage in decimal form.

For instance, if the discount amount is \)40 and the discount percentage is 20% (or 0.20 in decimal), you calculate \( \text{Original Price} = \frac{40}{0.20} \), which results in $200. Understanding this helps you evaluate deals and compare prices effectively.

Prealgebra Problems

Prealgebra is a fundamental stepping stone in mathematics that prepares you for more complex algebraic concepts. In dealing with discounts and original pricing, prealgebra helps in forming and solving simple equations. These problems typically involve basic arithmetic operations such as addition, subtraction, multiplication, and division. Prealgebra problems often, like in this case, require:

• Recognizing given information and unknowns
• Setting up equations based on relationships between quantities
• Using proportions and percentages to solve real-world problems

By practicing these types of exercises, you gain confidence in manipulating numbers and developing problem-solving skills. This foundation supports success in more advanced mathematics lessons later on.

Percentage Discount Problems

Percentage discount problems are practical math challenges, allowing you to apply percentages, a key concept in math, to everyday life. These problems generally ask you to find either the discount amount, the original price, or the sale price. Solving these problems can seem tricky at first, but it becomes easier with practice. Here's a structured approach:

• Identify the - Discount percentage - Discount amount or sale price
• Use the discount formula to set up an equation
• Solve the equation to find the missing value

Whether you're shopping for clothes, electronics, or groceries, understanding how percentage discounts work helps you make smarter financial choices and spot real savings in sales and promotions.