Problem 65 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 8+4 \div(3-1) $$

Step-by-Step Solution

Answer

The solution to the expression \(8+4 \div(3-1)\) is \(10\).

Step 1: Apply the Parentheses Rule

According to the order of operations, calculations inside parentheses (also known as brackets) should be performed first. Our expression includes brackets, so we first deal with the operation inside the brackets. The operation inside the parentheses is subtraction: \(3-1\). We perform this operation to give the result as \(2\).

Step 2: Perform Division

After handling the parentheses, the next operations to be performed, from left to right as per BIDMAS / PEMDAS, are multiplication and division. Our expression now reads as \(8+4 \div 2\). The division operation is \(4 \div 2\), so we perform this operation next and get the result as 2.

Step 3: Perform Addition

The last operation to be carried out as per the order of operations is the addition operation. The expression is now \(8 + 2\). Adding these two numbers yields the result as \(10\).

Key Concepts

Parentheses

When you come across parentheses (also known as brackets) in a mathematical expression, it's crucial to solve what’s inside them first. This rule is part of a broader principle called the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
The presence of parentheses indicates that these enclosed operations are prioritized. In our example expression, \(3-1\) is within the parentheses. Solving this subtraction first, we simplify it to \(2\). Think of parentheses as a "do me first" sign in any expression.

Parentheses are used to clarify operations and ensure operations are conducted in the correct order.

Always address operations inside parentheses as your first step.

Once solved, you can then focus on the rest of the expression with a now simplified equation.

Division

Once the operations inside the parentheses are complete, the next step is to scan for multiplication or division. According to the order of operations, once parentheses and exponents are solved, you handle multiplication and division from left to right. In our expression, \(8 + 4 \div 2\), the division, \(4 \div 2\), comes first before any addition.
Division helps break down numbers by determining how many times one number fits into another. In this scenario, dividing \(4\) by \(2\) results in \(2\). When you perform division correctly and at the right time, it greatly simplifies your overall calculation.

Division is sequential with multiplication, solved as you encounter them left to right.

It's crucial to use division to clarify and simplify complex expressions.

Addition

Addition is one of the last steps in the order of operations. It is performed after parentheses, exponents, multiplication, and division. In our expression, after solving the division, it simplifies to \(8 + 2\).
Addition is the process of bringing numbers together to find their total. Once other operations are resolved, addition combines what remains into a final simplified value. In this case, adding \(8\) and \(2\) gives a final result of \(10\).

Addition is simple yet follows after most other operations in order of execution.

Always add after sorting out any other possibilities such as multiplication or division first.

It ties together the remaining simplified figures to produce your final answer.