Problem 83 - Algebra 1: Concepts and Skills

Question

Simplify. \(\frac{27}{3}\)

Step-by-Step Solution

Answer

The simplified form of the fraction \(\frac{27}{3}\) is \(9\).

Step 1: Evaluate the Division

Divide the numerator by the denominator. That is \(27 ÷ 3\).

Step 2: Find the result

After dividing, we get \(9\).

Key Concepts

Division in Algebra

In algebra, division is an operation that tells us how many times a number, called the divisor, can be contained in another number, known as the dividend. The result of this operation is the quotient.

To simplify an expression involving division, like the problem \(\frac{27}{3}\), we divide the numerator (27) by the denominator (3). Division in algebra adheres to the same rules as arithmetic. In this example, 27 divided by 3 gives us a quotient of 9. This straightforward process is pivotal when we handle more complex algebraic expressions where variables are involved.

Numerical Expressions

A numerical expression is a mathematical sentence involving only numbers and operation symbols like plus (+), minus (-), times (\(*\)), and divided by (\(/\)). Numerical expressions do not include an equal sign. For instance, \(\frac{27}{3}\) is a numerical expression because it includes numbers and a division symbol.

To evaluate such expressions, we perform the operations in a specific order – called the order of operations. However, within this example, as we only have division, we proceed directly to simplifying it by carrying out the division. Understanding how to work with numerical expressions is fundamental in developing algebraic proficiency.

Solving Algebraic Expressions

Solving algebraic expressions involves finding the values of variables that make the expression true. Yet, when we deal with numerical expressions, no variables are present, and 'solving' simply means simplifying the expression down to a single number.

In the situation of \(\frac{27}{3}\), there are no variables, so we directly simplify the expression by performing the division. The principles of solving algebraic expressions scale up when variables are introduced; each operation provides a step closer to isolating the variable and determining its value.

Basic Algebraic Operations

Algebra is built on basic operations: addition, subtraction, multiplication, and division. These operations are the foundation of all algebraic manipulations.

When simplifying algebraic expressions, it's important to understand these basic operations not only with numbers but also how they interact with variables. For example, to simplify an expression like \(\frac{27}{3}\), we need only perform division. But if our expression included variables, we would apply the same basic operations while following additional algebraic rules, such as distribution or combining like terms. Grasping these basic operations is integral to advancing in algebra and tackling more complex problems.