Problem 12 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$8 \cdot 4 \div 16+5$$

Step-by-Step Solution

Answer

Answer: The correct value of the given expression is $$7$$.

Step 1: Identify the operations

In the given expression $$8 \cdot 4 \div 16+5$$, there are three operations: multiplication, division, and addition.

Step 2: Perform the multiplication and division

According to the order of operations, we must perform the multiplication and division first, from left to right. So, we will first calculate $$8 \cdot 4$$ and then divide the result by $$16$$. $$8 \cdot 4 = 32$$ Now, divide $$32$$ by $$16$$: $$32 \div 16 = 2$$

Step 3: Perform the addition

Now that we have the result of the multiplication and division, we can perform the addition: $$2 + 5 = 7$$

Step 4: Final answer

The correct value of the given expression, following the order of operations, is $$7$$.

Key Concepts

Understanding Multiplication

Multiplication is one of the fundamental operations in mathematics. It involves combining groups of equal sizes. When multiplying numbers, you are essentially adding a number to itself repeatedly.
For example, in the expression \(8 \cdot 4\), you multiply 8 by 4. This means you create a set of four groups, each containing 8 items. The result is 32, because when you add 8 four times, you get 32.

• Think of multiplication as a fast way of doing repeated addition.
• The symbol \(\cdot\) indicates multiplication.
• In terms of order, multiplication is typically performed before addition.

In any expression, multiplication should be addressed early based on precedence rules, unless parentheses indicate otherwise. Understanding multiplication helps tackle more complex equations smoothly.

Explaining Division

Division is the process of splitting a number into equal parts. It's the inverse operation of multiplication. In our example \(8 \cdot 4 \div 16 + 5\), the division part is \( 32 \div 16\).
Here, you take 32 and divide it into 16 equal parts. This calculation ultimately results in 2 because 16 fits into 32 exactly twice.

• View division as determining how many times one number fits into another.
• The division symbol is often represented by \(\div\).
• In expressions, perform division right after multiplication and before addition to maintain correct order of operations.

It's crucial to proceed with division after completing any multiplications, unless specified by parentheses. This orderly approach ensures accurate results.

Introducing Addition

Addition is one of the simplest arithmetic operations and involves combining numbers to get a sum. In our step-by-step solution, the final operation is adding 2 and 5 in the expression \(8 \cdot 4 \div 16 + 5\).
Once you have completed both multiplication and division, you proceed to addition:\(2 + 5 = 7\).

• Addition combines numbers or results from previous operations.
• The symbol \(+\) represents addition.
• Add last in the order of operations unless parentheses group them to be done first.

By understanding addition as the final step in expressions like these, it reveals the ultimate outcome. Respecting the sequence enables you to reach accurate solutions without errors.