Problem 89 - Prealgebra

Question

Use the rule for order of operations to simplify each expression. $$12+18 \div 2-1$$

Step-by-Step Solution

Answer

The expression simplifies to 20.

Step 1: Division

According to the order of operations, division and multiplication should be performed before addition and subtraction. Therefore, start by solving the division part of the expression: \(18 \div 2\). This equals \(9\). The expression now becomes \(12 + 9 - 1\).

Step 2: Addition

Next, perform addition. Adding \(12 + 9\) results in \(21\). The expression now becomes \(21 - 1\).

Step 3: Subtraction

Finally, perform the subtraction. Subtract \(1\) from \(21\) which results in \(20\). Therefore, the simplified expression is \(20\).

Key Concepts

Understanding Division

To simplify mathematical expressions involving several operations, such as division, it's important to follow specific rules, often referred to as "order of operations." Division is one of those key operations that must be addressed correctly within these rules. When you encounter a division in a longer mathematical expression—as in this exercise—it is typically handled after parentheses and exponents, but before any addition or subtraction. This ensures calculations are performed accurately.
In our exercise, the original expression is

• 18 divided by 2, which calculates directly to 9.

This division is crucial to simplifying the expression correctly. By addressing the division first, we maintain the integrity of the calculation and avoid errors in later steps. This step updates the problem to shorten and simplify it, setting us up for the next operations.

Applying Addition

Once any necessary divisions within an expression have been completed, moving onto addition comes next in the process. Addition combines values, bringing them together into a single result. In our example, we have already simplified the expression by performing the division, and we're left with

• 12 + 9 - 1.

At this point, our focus shifts to the addition aspect:

• Adding 12 and 9, we find the total is 21.

This operation is straightforward, serving to accumulate the results of prior calculations in the expression, leaving us with a new, simplified version to tackle in the final step.

Executing Subtraction

Subtraction is often the final step in simplifying expressions, providing the "clean up" to reach a final, simplified result. Subtraction involves finding the difference between numbers, reducing the total previously gathered. In this exercise, after performing both division and addition, our expression now looks like

• 21 - 1.

We perform the subtraction to find the final result:

• Subtracting 1 from 21 results in 20.

This operation completes the simplification process, following the full order of operations—division first, then addition, and lastly subtraction—to ensure the expression is fully simplified and correct. Understanding this order is key to mastering more complex mathematical tasks.