Question

You are shopping for school supplies. A store is offering a \(10 \%\) discount on binders and a \(20 \%\) discount on packages of paper. You want to buy 5 binders originally marked $$ 2.50\( each and 10 packages of paper originally marked $$ 1.30 each. a. Write an expression that shows how much you will save after the discounts. b. Evaluate the expression. c. Writing If you have $$ 25\) to spend on supplies, how much money will you have left over? Explain how you arrived at your answer.

Short answer

a) The total savings after the discounts will be $$3.85, b) The total cost for the school supplies after discount is $$21.65, c) The amount left after shopping from a budget of $$25 will be $$3.35.

Step-by-step solution

Calculate Discounts

First we need to calculate the amount of discount that will be applied to the binders and paper respectively. For binders, the discount is \(10 \% \) of $2.50, which is calculated as \(0.10 * 2.50 = $0.25\). For paper, the discount is \(20 \% \) of $1.30, which is calculated as \(0.20 * 1.30 = $0.26\).

Calculate Total Savings

After calculating each discount, lets find out the total savings. For binders, the total savings would be \(5 * 0.25 = $1.25\) and for papers it would be \(10 * 0.26 = $2.60\). Thus, the total savings will be \( $1.25 + $2.60 = $3.85\)

Calculate Total Cost With Discounts

Now, to calculate the total discounted cost of the items, first subtract the discount from original price and multiply by the quantity. So, for binders it would be \((2.50 - 0.25) * 5 = $11.25\) and for paper it would be \((1.30 - 0.26) * 10 = 10.40\). So, total cost to be paid is \(11.25 + 10.40 = $21.65\).

Calculate Remaining Money

The last step is to calculate how much money will be left after the purchase. This is done by subtracting the total cost from the initial budget, which is \(25 - 21.65 = $3.35\). Therefore, $3.35 will be left over after the shopping.

Key concepts

Percent Discount

Understanding a percent discount is crucial when calculating how much money you can save on a purchase during a sale. A percent discount refers to the reduction in price of an item by a certain percentage. For instance, if an item is priced at \(100 and a store offers a 20% discount, the price reduction would be \)20. The formula to calculate a discount is the original price times the discount rate.

To apply the formula, first convert the percentage into decimal form, which is done by dividing the percentage by 100. In the provided example, the discounts are 10% for binders and 20% for packages of paper. Therefore, the discount rates are 0.10 and 0.20 in decimal form respectively. To calculate the savings for multiple items, multiply the discount for one item by the number of items you intend to buy.

Writing Algebraic Expressions

Writing algebraic expressions is a foundational skill in algebra. It involves translating a verbal description or real-world situation into mathematical symbols and numbers that can represent a particular arithmetic relationship or operation. The benefit of an algebraic expression is that it allows us to perform calculations systematically and understand the relationships between quantities.

In our context, writing an algebraic expression to describe the discounts on school supplies requires identifying both the quantities involved and their relationships. For example, if 'x' represents the original price of a binder, and the discount rate is 10%, the expression for the discount on one binder would be `0.10 * x`. For five binders, we'd write `5 * (0.10 * x)`. Properly written algebraic expressions are succinct and avoid ambiguity, which makes calculating discounts much simpler and clearer.

Evaluating Expressions

Evaluating expressions is the process of substituting numbers for variables in an algebraic expression and then simplifying to get a numerical result. It's essential to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition, and Subtraction), to accurately calculate the value of the expression.

For example, to calculate the total savings after discount for binders and paper in the previous steps, we multiply the discount per item by the number of items and then add up these amounts for both binders and paper. This gives us the expression `5 * (0.10 * 2.50) + 10 * (0.20 * 1.30)` which simplifies to `5 * 0.25 + 10 * 0.26`, further simplifying to $3.85. This result represents the total savings from the discounts, which is determined by evaluating the expression accurately.