Problem 14 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$3(8+2) \div 6+3$$

Step-by-Step Solution

Answer

Answer: 8

Step 1: Solve expressions inside Parentheses

First, we need to find the sum inside the parentheses: $$3(8+2) \div 6+3$$ $$(8+2)=10$$ So, the expression becomes: $$3\cdot 10 \div 6+3$$

Step 2: Perform Multiplication and Division

Next, we'll perform multiplication and division operations from left to right. $$3\cdot 10 \div 6+3$$ Multiply 3 and 10: $$30 \div 6+3$$ Now, divide 30 by 6: $$5+3$$

Step 3: Perform Addition

Lastly, perform addition: $$5+3=8$$ Therefore, the value of the expression $$3(8+2) \div 6+3$$ is 8.

Key Concepts

Parentheses

Parentheses are an integral part of the order of operations. They allow us to specify which operations should be carried out first in a mathematical expression. By enclosing certain calculations within parentheses, we signal that these steps must be completed before any other operations can take place. This is crucial in ensuring that we arrive at the correct answer.

In the expression given, 3(8+2) \div 6 + 3, the numbers inside the parentheses—8 and 2—are treated as a separate entity. We calculate the sum inside the parentheses first:

• Calculate: \(8 + 2\)
• The result is: \(10\)

Substituting back into the expression gives us a new simplified form: \(3 \cdot 10 \div 6 + 3\). This highlights the importance of parentheses: they ensure that certain parts of a calculation are completed before moving on to the next operations.

Multiplication and Division

Once we've addressed any operations inside parentheses, the next step in order of operations is multiplication and division. When these operations are part of an expression, they are carried out in sequence from left to right, as they appear.

In our given expression, \(3 \cdot 10 \div 6 + 3\), after resolving the parentheses, multiplication and division must be addressed before tackling any addition or subtraction. Here's how it goes:

• First, multiply: \(3 \cdot 10 = 30\)
• Next, divide: \(30 \div 6 = 5\)

The sequence is important; starting with multiplication first, followed by division, aligns with the natural left-to-right progression in mathematical operations. Thus, by tackling this step correctly, you're ensuring the operation is performed properly.

Addition and Subtraction

After dealing with parentheses, multiplication, and division, we are left with handling addition and subtraction. These operations are also processed from left to right.

In our specific problem, once we've performed all necessary multiplications and divisions, we're left with the expression \(5 + 3\). At this stage, it's straightforward:

• Add: \(5 + 3 = 8\)

Addition and subtraction typically come after higher-priority operations like multiplication and division, which is why they're executed last in this step-by-step approach. This method of following the order of operations ensures accuracy in mathematical problem-solving by maintaining a consistent calculation procedure.