Problem 47 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$-100 \div 10 \div 2$$

Step-by-Step Solution

Answer

The simplified expression is -5.

Step 1: Identify the Operations

The given expression is \[-100 \div 10 \div 2\]It involves repeated division.

Step 2: Apply Division Sequentially

Start with the leftmost division operation as per the order of operations rules (PEMDAS/BODMAS), which state to perform operations from left to right when they are of the same precedence. Divide \[-100 \div 10\]This equals \[-10\].

Step 3: Divide the Result Again

Now, take the result from Step 2 and perform the next division operation:\[-10 \div 2\]This results in \[-5\].

Key Concepts

The Division Process Explained

Division is a basic arithmetic operation that simplifies numbers by determining how many times one number is contained within another. In the expression \(-100 \div 10 \div 2\), we perform division in a left-to-right sequence since division is an operation of equal precedence when in a sequence.

To solve \(-100 \div 10\), think about how many times 10 fits into -100. The answer is -10, as ten fits into one hundred ten times, and the negative sign of one hundred flips the sign of the result.
Next, we divide this result by 2. Take the \(-10\) from the first division and divide it by 2 to get \(-5\). This is because 2 goes into 10 five times. Again, the sign is negative as positive and negative together result in a negative (\(+ \times - = -\)).

Understanding PEMDAS

PEMDAS is an acronym for remembering the order of operations used in mathematics to solve expressions and equations. It helps you know which operation to perform first in cases where there's a mix of different operations. The letters stand for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

When solving an expression with multiple operations, start by solving anything inside parentheses first. Then, address any exponents. After these, move on to multiplication or division, working from left to right. Finally, tackle addition or subtraction, also from left to right.

It's crucial to perform operations with the same rank, like multiplication and division, from left to right. This ensures calculations are performed correctly and consistently, leading to the correct result.

Grasping BODMAS

Just like PEMDAS, BODMAS is a mnemonic to help remember the order in which to tackle different parts of mathematical problems. The acronym BODMAS stands for:

• B: Brackets
• O: Orders (another term for exponents)
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction

BODMAS functions similarly to PEMDAS and serves to remind us to first handle the calculations inside brackets (or parentheses), then any orders or exponents. As division and multiplication have the same level of importance, they should be performed as they appear from left to right. The same rule applies to addition and subtraction.
This method ensures calculations are done systematically, helping avoid mistakes that arise from performing operations out of order.