Solving the Expression 18 x 12 – 4
This article provides a comprehensive breakdown of how to solve the arithmetic expression 18 x 12 – 4, explaining the order of operations, alternative representations, real-world applications, and extensions to more complex problems.
Understanding the Expression
Solving the expression 18 x 12 – 4 requires understanding the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms dictate that multiplication and division are performed before addition and subtraction.
Here’s a step-by-step solution:
- Multiplication: First, we perform the multiplication: 18 x 12 = 216.
- Subtraction: Next, we perform the subtraction: 216 – 4 = 212.
Therefore, the solution to 18 x 12 – 4 is 212.
Alternative methods include using a calculator or breaking down the multiplication into smaller, easier steps (e.g., 18 x 10 + 18 x 2).
A visual representation of the step-by-step process:
18 x 12 - 4 ↓ 216 - 4 ↓ 212
Alternative Representations
The order of operations can be explicitly shown using parentheses. The expression remains equivalent if we write it as (18 x 12) – 4.
Equivalent expressions that yield the same result can be created using the associative and commutative properties of addition and multiplication. However, it’s important to note that these properties apply to addition and multiplication separately; they cannot be directly applied to mix addition and multiplication without altering the result.
Here’s a table illustrating different equivalent expressions (note that creating truly “different” equivalent expressions without parentheses is limited in this case):
Expression | Equivalent Expression | Result -----------------------|-----------------------|-------- 18 x 12 - 4 | (18 x 12) - 4 | 212 18 x 12 - 4 | (18 x 10) + (18 x 2) -4 | 212
Applications and Context
Calculations like 18 x 12 – 4 are commonly used in various real-world scenarios. For example, calculating the total cost of 12 items priced at $18 each, then subtracting a $4 discount.
Comparing this expression to similar ones:
- 18 x (12 – 4): This expression, following the order of operations, becomes 18 x 8 = 144. The parentheses change the order of operations significantly.
- 18 – 4 x 12: This equals 18 – 48 = -30. This highlights the importance of adhering to the order of operations.
A common error is performing the subtraction before the multiplication. To avoid this, always remember the PEMDAS/BODMAS rule.
Accuracy is crucial in such calculations; an error can lead to incorrect results in various contexts, from financial transactions to engineering calculations.
Extending the Concept
A more complex problem could be: 25 x (36 + 14) – 72 ÷ 8 + 15.
Solution:
- Parentheses: 36 + 14 = 50
- Multiplication: 25 x 50 = 1250
- Division: 72 ÷ 8 = 9
- Subtraction: 1250 – 9 = 1241
- Addition: 1241 + 15 = 1256
The solution is 1256.
A word problem: A baker makes 18 batches of cookies, each containing 12 cookies. If 4 cookies are accidentally burned, how many cookies are left?
The concepts involved in solving 18 x 12 – 4 are fundamental to algebra and more advanced mathematical topics such as solving equations and working with polynomials.
