Problem 102 - Algebra for College Students

Question

The following simplification problem is incorrect. The answer should be \(-11\). Find and correct the error. $$ \begin{aligned} 8 \div(-4)(2)-3(4) \div 2+(-1) &=(-2)(2)-12 \div 1 \\ &=-4-12 \\ &=-16 \end{aligned} $$

Step-by-Step Solution

Answer

The corrected simplification is \(-11\).

Step 1: Simplify Division and Multiplication

We begin by addressing the operations in the expression according to the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Division and Multiplication, Addition and Subtraction). The expression given is:\[8 \div(-4)(2)-3(4) \div 2+(-1)\]First, we handle the division and multiplications from left to right:- For \(8 \div (-4)\), we get \(-2\).- Then \((-2)(2) = -4\).- For \(-3(4) \div 2\), treat this as a separate group: first, calculate \(-3 \times 4 = -12\), then \(-12 \div 2 = -6\).Thus, rewriting the expression with intermediate results:\[-4 - 6 + (-1)\]

Step 2: Simplify Addition and Subtraction

Now, we evaluate the simplified expression:\[-4 - 6 + (-1)\]The expression can further be simplified as:- First calculate \(-4 - 6\), which gives \(-10\).- Then add \(-1\) to \(-10\), resulting in \(-11\).Thus, the final simplified form of the initial expression is \(-11\).

Key Concepts

Simplification of Expressions

Simplification of expressions often involves breaking down an expression step by step, making it easier to solve. The key is to follow the correct order of operations, commonly remembered by the acronym PEMDAS/BODMAS. This stands for Parentheses/Brackets, Exponents/Orders, Division and Multiplication, and Addition and Subtraction. By adhering to this order, you simplify complex mathematical expressions accurately. In our exercise, simplification begins with identifying and appropriately managing different operations: division and multiplication are prioritized over addition and subtraction. Mistakes often occur when the operations are not carried out in the correct sequence or if a negative sign is mismanaged. Throughout this simplification process, changing the expression's format by resolving operations correctly can lead to a simpler, more understandable result.

Division and Multiplication

When tackling division and multiplication within an expression, it is essential to handle these operations from left to right. This is because both division and multiplication have equal precedence in the order of operations. For instance, in our example, we began with the expression:\[8 \div (-4)(2) - 3(4) \div 2 + (-1)\]Here's what we did:- **Divide the numbers first**: Take the first operation, \(8 \div (-4)\). Carrying out this division gives \(-2\).- **Multiply the result**: Follow with \((-2) \times 2\), resulting in \(-4\).- **Handle grouped multiplication and division**: Treat \(-3(4) \div 2\) as a batch of operations, starting with multiplication: \(-3 \times 4 = -12\), followed by division: \(-12 \div 2 = -6\).Remember, performing multiplication or division out of order can lead to errors, as each step builds on the previous results.

Addition and Subtraction

Once division and multiplication are complete, the next step is to simplify the expression through addition and subtraction. These operations are crucial because they adjust the value of the expression in the final stages.In the simplified expression:\[-4 - 6 + (-1)\]The process was as follows:- **Perform subtraction first**: Start with \(-4 - 6\). Here, you subtract 6 from \(-4\), resulting in \(-10\).- **Incorporate addition or further subtraction**: Finally, combine \(-10\) with \(-1\), leading to \(-11\).By managing each operation carefully, avoiding confusion with negative numbers, and using correct arithmetic, the expression simplifies effectively. This results in a clear and precise solution, showcasing the importance of orderly addition and subtraction after division and multiplication.