Problem 28 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 4 \cdot 2+15 \div 3 $$

Step-by-Step Solution

Answer

The evaluated expression is 13.

Step 1: Perform Multiplication and Division Operations First

According to the order of operations (PEMDAS/BODMAS), multiplication and division should be performed before addition and subtraction. So, \(4 \cdot 2 = 8\) and \(15 \div 3 = 5\).

Step 2: Perform Addition

Next, perform the addition operation \(8 + 5 = 13\)

Key Concepts

PEMDAS

Understanding the correct sequence of mathematical operations is crucial in solving algebraic expressions. PEMDAS is an acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule helps students remember the order in which they should perform operations to accurately evaluate mathematical expressions.

For instance, when faced with an expression such as \(4 \cdot 2 + 15 \div 3\), following PEMDAS, we tackle Multiplication and Division before Addition. So, \(4 \cdot 2 = 8\) and \(15 \div 3 = 5\) are calculated first, ensuring that operations within the expression are performed correctly according to the hierarchy prescribed by PEMDAS.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They're a fundamental component of algebra that students must learn to manipulate and evaluate correctly. Unlike simple arithmetic, where we only have numbers, algebra introduces letters that represent unknown quantities or variables that can take various numerical values.

To simplify or solve algebraic expressions, one must apply the order of operations appropriately. In expressions like our example \(4 \cdot 2 + 15 \div 3\), even though there are no variables, treating numbers with the same principles applied in algebraic manipulation ensures consistency and helps build foundational skills for more advanced algebraic problem solving.

Basic Arithmetic

Basic arithmetic includes operations such as addition, subtraction, multiplication, and division. These are the building blocks for more complex mathematical problem-solving. By mastering basic arithmetic, students can approach algebraic expressions with confidence. In the expression \(4 \cdot 2 + 15 \div 3\), it's essential first to do multiplication and division, as they take priority over addition and subtraction.

This hierarchy is not arbitrary; multiplication and division are fundamentally repeated addition and subtraction, respectively. Therefore, it's logical to simplify those operations first to avoid errors in the overall calculation. With practice and understanding, applying these rules becomes second nature.

BODMAS

BODMAS is another acronym similar to PEMDAS, and it stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). This rule is prevalent in some regions and serves the same purpose as PEMDAS, guiding the sequence of operations to ensure mathematical accuracy.

Applying BODMAS to our example expression, Division and Multiplication are prioritized: \(15 \div 3 = 5\) and \(4 \cdot 2 = 8\), followed by Addition to combine the results: \(8 + 5 = 13\). It's key for students to understand that both PEMDAS and BODMAS lead to the same solution, as they are simply different ways to remember the same underlying order of operations.