Problem 76
Question
Answer with an algebraic expression. See Example 9. A certain type of office desk that used to sell for \(\$ x\) is now on sale for \(\$ 50\) off. What will a company pay if it purchases 80 of the desks?
Step-by-Step Solution
Verified Answer
The company will pay \\(80x - 4000\) dollars.
1Step 1: Understand the Problem
We need to find the total cost for a company to purchase 80 desks, each of which is now \\(50 less than its original price of \\) x. The desks are on sale, so the company will save money on each desk.
2Step 2: Calculate the Discounted Price of One Desk
The original price of one desk is \\( x. With a \\)50 discount, the new price of one desk is \(x - 50)\) dollars.
3Step 3: Find the Total Cost for 80 Desks
The total cost for purchasing 80 desks at the discounted price is obtained by multiplying the price of one desk by 80. This results in \(80 \times (x - 50)\).
4Step 4: Simplify the Algebraic Expression
Use the distributive property to simplify the expression: \[80 \times (x - 50) = 80x - 4000\].
Key Concepts
Understanding the Distributive PropertyHow to Calculate DiscountsDetermining Total Costs
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows for the simplification of expressions. It provides a way to multiply a single term across terms within parentheses. Imagine you have an expression like \(a(b + c)\). The distributive property helps you break this down into \(ab + ac\). In essence, you're distributing the term outside the parentheses to each term inside.
In the original problem, to determine the total cost for 80 desks, we applied the distributive property: \[80(x - 50) = 80x - 4000.\]This breakdown enables us to easily manage larger calculations and see the impact of the discount clearly by multiplying each part separately.
- Mental Easy: Think of it like sharing something equally. If you distribute 80 chocolates into 2 boxes of size \(x\) and size \(50\), each box would get some chocolates.
- Simplifying: Make algebraic expressions cleaner and easier to work with.
- Examples: Distributing knowledge is like distributing chocolates!
In the original problem, to determine the total cost for 80 desks, we applied the distributive property: \[80(x - 50) = 80x - 4000.\]This breakdown enables us to easily manage larger calculations and see the impact of the discount clearly by multiplying each part separately.
How to Calculate Discounts
A discount refers to the amount subtracted from the original price. It's what makes shopping fun! Understanding discounts can also make budgeting easier and helps in saving money.
By mastering discount calculations, individuals and businesses can expertly manage their finances. In our problem, this meant calculating \[x - 50\]per desk to determine the reduced rate.
- Finding the Discount: The initial task is to determine the dollar amount of the discount. For instance, in our desk scenario, each desk originally costs \(x\) but now is offered for \(\$50\) less.
- Discounted Price: This is obtained by subtracting the discount from the original price. So, the new reduced price for one desk is \(x - 50\).
- Benefits of Discounts: Businesses use them to encourage purchases; shoppers use them to get more for less.
By mastering discount calculations, individuals and businesses can expertly manage their finances. In our problem, this meant calculating \[x - 50\]per desk to determine the reduced rate.
Determining Total Costs
Once you've grasped how to simplify costs using the distributive property and identified discounted prices, the next step is to calculate total expenses. Whether you're in business or just organizing a party, this is crucial for budgeting.
Using such calculations, businesses can efficiently plan, ensuring that when desk supplies are bought in bulk, they are done economically, reflecting calculated decisions. This aids in managing large-scale purchases with minimal errors.
- Gather Information: Know the number of items and their price. Here, it was about 80 desks, each reduced by \(\$50\).
- Apply Your Knowledge: Calculate the total cost using multiplication. We found the price for one desk (\(x - 50\)) and multiplied by 80, representing the total number of desks purchased.
- Usage of Expression: The expression \(80x - 4000\) neatly represents total costs, showing both the gross value based on original price and the reduction made by the discount.
Using such calculations, businesses can efficiently plan, ensuring that when desk supplies are bought in bulk, they are done economically, reflecting calculated decisions. This aids in managing large-scale purchases with minimal errors.