Problem 49 - Elementary Algebra

Question

For the following problems, rewrite each phrase using algebraic notation. $$(a+b) \text { divided by }(a+4)$$

Step-by-Step Solution

Answer

#Short Answer# The algebraic notation of the given expression is $${(a+b) \over (a+4)}$$.

Step 1: Identify the terms and the operator

In the given expression, there are two terms (a+b) and (a+4), and we need to divide them.

Step 2: Replace words with mathematical symbols

To rewrite this expression using algebraic notation, replace the words "divided by" with a fraction symbol. This will represent the division operation. So, the expression becomes: $${(a+b) \over (a+4)}$$

Key Concepts

Understanding Expressions in Algebra

Expressions in algebra are a combination of numbers, variables, and operations. They describe a calculation without having to solve it immediately. In the context of our exercise, the expressions are \(a + b\) and \(a + 4\). Here, \(a\) and \(b\) are variables, which means they can represent any number. The symbols \(+\) and \(4\) are constants and arithmetic operators respectively.
Each part of an algebraic expression serves a purpose:

• Variables represent unknown values or quantities.
• Constants are fixed numbers that don’t change.
• Operators like \(+\), \(-\), \(\times\), or \(\div\) dictate how the terms are combined.

Understanding expressions is the first step in performing operations on them, such as addition or division.

The Role of Fractions in Expressions

Fractions are a way to represent division in algebraic expressions. They consist of two parts: a numerator, which appears above the division line, and a denominator, which appears below. In our exercise expression \(\frac{a+b}{a+4}\), \(a + b\) is the numerator and \(a + 4\) is the denominator.
Fractions are useful because:

• They help see that one value is divided by another.
• They keep complex expressions organized.
• They simplify problems when rearranging or solving equations.

The key is to remember that the fraction bar itself can be read as "divided by," clarifying the relationship between the expression components.

Division in Algebraic Notation

Division is one of the fundamental operations in algebra that can be represented in different ways, but is often shown using fractions. In our expression \(\frac{a+b}{a+4}\), division is represented by the fraction bar. The phrase “divided by” informs us of how the division should be set up.
Key points about division in algebra include:

• Division splits a value into parts as shown by a fraction.
• Understanding the direction of division is critical. Usually, the numerator is divided by the denominator.
• In algebra, division can also involve simplifying expressions or solving equations.

By using algebraic notation, division is easily represented and manipulated within equations and expressions.