Problem 60 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$ 6(8.75) $$

Step-by-Step Solution

Answer

The simplified expression is \(52.5\).

Step 1: Identifying the distributive property

The distributive property in math is an algebra property that is used when an operation is performed on a number within parentheses. To apply this property in our example, the number \(6\) needs to be multiplied with each of the number inside the parentheses. This would be expressed as follows: \(6 \times 8.75 = 6 \times (8 + 0.75)\).

Step 2: Distribute the multiplication

Now, we can apply the distributive property to make the calculation easier. We multiply \(6\) with each term inside the parentheses separately: \(6 \times 8 + 6 \times 0.75 = 48 + 4.5\).

Step 3: Add the Results

The final step is to add the two results from our separate calculations. Adding those two numbers results in the final answer: \(48 + 4.5 = 52.5\).

Key Concepts

Mental Math

Mental math refers to the practice of performing arithmetic calculations without the aid of tools like calculators or pencil and paper. It's an essential skill that not only speeds up the process of mathematical computation but also enhances cognitive abilities, such as memory and concentration.

When it comes to using mental math to simplify expressions, strategies such as breaking down complex numbers into smaller, more manageable parts or recognizing patterns can be incredibly helpful. For example, in the exercise \(6(8.75)\), mental math would involve seeing \(8.75\) as \(8 + 0.75\) to simplify the process of multiplying by \(6\). Instead of calculating the entire product at once, the number \(8\) and \(0.75\), being simpler numbers, are individually multiplied by \(6\). This approach transforms a potentially complicated multiplication into smaller, quicker additions.

Another useful mental math technique is estimation, where one rounds off numbers to make the arithmetic easier, then adjusts the answer for the initial round-off. In the exercise above, instead of using \(0.75\), estimating it to \(1\) could have been an alternative (though less precise) mental math strategy.

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra that involves reducing an expression to its simplest form, making it easier to understand and work with. The process can include combining like terms, factoring, expanding expressions, or applying mathematical properties such as the distributive property.

To simplify an expression, one might start by applying the distributive property, as seen in the example \(6(8.75)\). Here, the expression inside the parentheses is not easily multiplied by \(6\), so it's decomposed into two terms, \(8\) and \(0.75\). Once simplified using the distributive property, we end up with \(6 \times 8 + 6 \times 0.75\), which is far less daunting.

The art of simplifying expressions is crucial not only for solving mathematical problems efficiently but also for understanding more complex mathematical concepts. As expressions become more simplified, patterns can be noticed, hypotheses developed, and generalizations formed, which are all key to the study of mathematics.

Algebraic Properties

Algebraic properties are rules that are universally true in mathematics and are applied to manipulate expressions and equations. They are the backbone of algebra and include properties like the distributive property, associative property, commutative property, and identity property, among others.

The distributive property, used in the exercise \(6(8.75)\), is expressed as \(a(b + c) = ab + ac\). It states that a number outside the parentheses can be distributed to each term within the parentheses and then added together. This property is invaluable when simplifying algebraic expressions and solving equations because it streamlines computations and unravels complex expressions into more workable forms.

Understanding and properly applying algebraic properties like the distributive property enables students to progress through more intricate algebraic problems with confidence. These properties are not just mere ‘rules’ to follow; they reveal the structural relationships within math that are consistent and reliable, forming the very foundation of every mathematical operation.