Question

Evaluate each expression. \(\frac{1}{2}(26+20)\)

Short answer

The expression evaluates to 23.

Step-by-step solution

Evaluate the Expression Inside Parentheses

The first step is to evaluate the expression inside the parentheses. We have \(26 + 20\). Adding these two numbers together gives us \(46\). So, we replace \((26 + 20)\) with \(46\).

Multiply by the Fraction Outside the Parentheses

Now that we have simplified the expression inside the parentheses to \(46\), we multiply by the fraction \(\frac{1}{2}\) outside the parentheses. This means we multiply \(\frac{1}{2}\) by \(46\), which is essentially finding half of \(46\).

Simplify the Resulting Expression

Calculate half of \(46\) by multiplying: \(\frac{1}{2} \times 46 = 23\). Therefore, after performing the multiplication, we are left with \(23\).

Key concepts

Understanding Order of Operations

When dealing with mathematical expressions, the order of operations is a crucial concept to grasp. It ensures that everyone solves expressions consistently and correctly. The standard sequence is often remembered through the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our given example, \(\frac{1}{2}(26+20)\), the expression inside the parentheses must be addressed first.
After handling what's in the parentheses, the multiplication by a fraction is next. Following this order ensures correct and logical steps to find the solution. Misunderstanding or skipping any step could lead to incorrect answers. It's always a good practice to simplify expressions step by step, according to these rules.

Evaluating Mathematical Expressions

Evaluating expressions involves simplifying them into their simplest form. This is typically done by performing arithmetic operations either given directly or indicated implicitly by bracketed sections. The goal is to resolve every component of the expression to find its equivalent value.
For expressions like \((26+20)\), we start by finding the sum inside the parentheses, resulting in a manageable number: 46.
After resolving the parentheses, any operations outside, such as multiplication by fractions, can proceed. In this example, multiplying 46 by \(\frac{1}{2}\) means finding half of 46.

• Calculate \((26 + 20) = 46\)
• Find half: \(46 \times \frac{1}{2}\)
• The result is 23

By evaluating expressions methodically, one can avoid errors and ensure precision in results.

Basic Arithmetic Operations

Arithmetic operations are the fundamental operations that form the basis of mathematics: addition, subtraction, multiplication, and division. Understanding how these operations interact is vital for evaluating expressions correctly.
In our example, we start with addition inside the parentheses:

• Addition: Combine 26 and 20 to get 46.

After handling addition, we move on to multiplication which uses properties of fractions:

• Multiplication with Fractions: Half of a number \(n\) is found using \(n \times \frac{1}{2}\).

This step gives us our final result, 23, after simplifying the expression. Becoming adept at these basic operations is essential for tackling more complex problems in mathematics with confidence.