Problem 38 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lesson 1.3) $$ 6+9 \div 3+3 $$

Step-by-Step Solution

Answer

The evaluation of the expression 6 + 9 \div 3 + 3 equals 12.

Step 1: Perform Division before Addition

According to the order of operations, division comes before addition. Therefore, the division operation, 9 \div 3, should be performed first: 6 + (9 \div 3) + 3. Calculating 9 divided by 3 equals 3.

Step 2: Perform Addition

Then, add the results together: 6 + 3 + 3. Firstly, add 6 and 3 gives us 9, then adding 9 and 3 gives us 12.

Key Concepts

Division

Division is one of the four basic operations in mathematics, alongside addition, subtraction, and multiplication. It involves splitting a number into equal parts. In our exercise, we have the expression \( 6 + 9 \div 3 + 3 \). Here, the division operation \( 9 \div 3 \) is the first to be addressed.
This is because, according to the mathematical order of operations, division (and multiplication) must be completed before addition (and subtraction).
To perform the division in this expression:

• Identify the numbers involved: 9 and 3.
• Calculate how many times 3 can be multiplied to make 9.
• The answer is 3, as 3 times 3 equals 9.

Thus, \(9 \div 3 = 3\). With this calculated, the expression simplifies to \(6 + 3 + 3\), ready for the next operation.

Addition

Addition is another crucial mathematical operation. It brings numbers together to form a total or sum. After completing the division in our expression and simplifying it to \(6 + 3 + 3\), we next focus on adding the numbers.
It's always good to tackle additions in a step-by-step manner:

• First, add 6 and 3. This results in 9.
• Next, add the result (9) to the remaining 3.
• Thus, the final total is 12.

Addition is largely associative, which means the order of adding numbers doesn't affect the total. Whether we calculate \((6 + 3) + 3\) or \(6 + (3 + 3)\), both will give us 12.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operators (such as plus or minus), and sometimes variables. They represent a particular value or set of operations to perform. In our case, the expression is \(6 + 9 \div 3 + 3\).
Understanding how to evaluate an expression correctly is key to solving mathematical problems. For this, we rely on the order of operations, often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Mathematical expressions can vary greatly in complexity:

• Simple, with just a couple of numbers and operations.
• Complex, with many operations, parentheses, and possibly exponents or variables.

It's crucial to perform the operations as per the given order to achieve the correct result, ensuring clarity and accuracy when evaluating expressions.