Problem 17 - Elementary Algebra Within Reach

Question

You are shopping for school supplies. You want to buy 8 notebooks for \(\$ 1.25\) each. Show how you can use the Distributive Property to find the total of the notebooks mentally.

Step-by-Step Solution

Answer

The total cost of the notebooks is $10.00

Step 1: Express the Cost of a Notebook as a Sum

First, express $1.25, the cost of one notebook, as the sum of $1.00 and $0.25. That is, $1.25 = $1.00 + $0.25.

Step 2: Apply the Distributive Property

Next, apply the Distributive Property of multiplication to solve for the total cost. This implies that the total cost of 8 notebooks is equal to \(8 * ($1.00 + $0.25)\) which simplifies to \(8 * $1.00 + 8 * $0.25\).

Step 3: Solve for Total Cost

Finally, solve the above simple multiplicative operations. That is, calculate \(8 * $1.00\) and \(8 * $0.25\) , and then add the results. The solution will give us the total cost of the 8 notebooks.

Key Concepts

Mental Math Techniques

Using mental math techniques can greatly simplify calculations, especially in shopping scenarios like buying multiple items. These tactics help us break down more complex calculations into smaller, easily manageable parts. By doing so, we can avoid the need for calculators and perform math quickly.

One effective mental math technique involves

• Breaking Numbers into Simpler Parts: This involves expressing a number as a sum of its simpler components.
• Rearranging Terms: This makes the numbers more comfortable to work with.

In the case of buying notebooks, the cost of one notebook, $1.25, is expressed as $1.00 + $0.25. By breaking it down this way, we can distribute the multiplication across these parts, making it easier to handle without writing anything down. This is applied further when you mentally multiply 8 notebooks by breaking down the operations.

Multiplication

Multiplication is one of the core operations in mathematics and a crucial tool in computing repeated additions efficiently. When applying multiplication in mental math, it often helps to visualize or compute simple parts separately and then combine them.

For the exercise, the task was to calculate the total cost of 8 notebooks costing $1.25 each. By using multiplication, you can quickly find how much all the notebooks cost together.

The initial equation appears as

• Multiply the total number of notebooks by the cost of each: \(8 \times (1.00 + 0.25)\).
• Apply multiplication to each term separately: \(8 \times 1.00\) and \(8 \times 0.25\).
• These operations give us intermediate values, which are then added: 8 dollars and 2 dollars.

The multiplication process simplifies the addition of repeated numbers, hence making it efficient to compute even large quantities mentally.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operations. They are used to represent relationships and help us perform calculations symbolically.

The Distributive Property is an essential tool that works within the context of algebraic expressions. It allows you to multiply a single term by all terms in a parenthesis, simplifying expressions to perform calculations more straightforwardly.

For instance, take the expression for calculating notebook costs:

• The expression begins as \(8 \times (1.00 + 0.25)\).
• By applying the Distributive Property, the expression becomes \((8 \times 1.00) + (8 \times 0.25)\).
• This allows easy multiplication of each separate term before summing the results: first multiplying 8 by 1.00, then by 0.25.
• The final, simplified expression gives us the overall cost of all items purchased.

Using algebraic expressions and understanding properties like the Distributive Property can greatly enhance your ability to tackle everyday mathematical problems with confidence.