Problem 103 - Intermediate Algebra

Question

Evaluate each expression. a. \(100-20+5\) b. \(100-(20+5)\) c. \(100 \div 20 \cdot 5\) d. \(100 \div(20 \cdot 5)\)

Step-by-Step Solution

Answer

a. 85, b. 75, c. 25, d. 1

Step 1: Solve Expression a

Evaluate the expression \(100-20+5\). Start by solving the subtraction: \(100 - 20 = 80\). Then, add 5 to the result: \(80 + 5 = 85\).

Step 2: Solve Expression b

Evaluate the expression \(100 - (20 + 5)\). Start by solving the expression inside the parentheses: \(20 + 5 = 25\). Then subtract 25 from 100: \(100 - 25 = 75\).

Step 3: Solve Expression c

Evaluate the expression \(100 \div 20 \cdot 5\). Division and multiplication are performed from left to right. First, divide 100 by 20: \(100 \div 20 = 5\). Then, multiply the result by 5: \(5 \cdot 5 = 25\).

Step 4: Solve Expression d

Evaluate the expression \(100 \div (20 \cdot 5)\). Start by solving the expression inside the parentheses: \(20 \cdot 5 = 100\). Then, divide 100 by 100: \(100 \div 100 = 1\).

Key Concepts

Expression Evaluation

Expression evaluation follows a specific set of rules known as the Order of Operations. These rules ensure that everyone gets the same result when solving mathematical expressions. The main tools in our toolbox are the arithmetic operations: addition, subtraction, multiplication, and division. Knowing the order to apply them is crucial. Generally, we follow the PEMDAS/BODMAS rule:

• P/B - Parentheses/Brackets
• E/O - Exponents/Orders
• MD/DM - Multiplication and Division (left to right)
• AS - Addition and Subtraction (left to right)

When evaluating expressions, always start with operations inside parentheses, then proceed according to the hierarchy mentioned above. This systematic process, known as 'expression evaluation,' ensures a structured approach to solving complex mathematical problems.

Arithmetic Operations

Arithmetic operations are the basic math processes. They include addition, subtraction, multiplication, and division. Understanding how to apply these operations correctly and in the correct order is key to mastering expression evaluation. Here's a short overview:

• Addition: Combines numbers to form a greater number.
• Subtraction: Finds the difference between numbers, reducing one by another.
• Multiplication: Increases a number by a factor of another, like repeated addition.
• Division: Splits a number into equal parts or groups.

Each of these operations plays a vital role in solving an expression. When multiple operations appear in one expression, remember to follow the order of operations, performing multiplication and division first (from left to right) and then addition and subtraction (also from left to right). This logical ordering is what ensures accurate solutions in arithmetic operations.

Parentheses in Calculations

Parentheses can greatly change the outcome of an expression by grouping certain calculations together. This means what's inside the parentheses has the highest priority in the order of operations. For example, consider this difference:

• The expression without parentheses: \[ 100 - 20 + 5 \] You perform subtraction followed by addition, resulting in \[85\].
• The expression with parentheses: \[ 100 - (20 + 5) \] Here, you first add inside the brackets to get \[25\], then subtract to obtain \[75\].

Thus, parentheses ensure that the grouped expression is solved first, which can dramatically affect the result. Always pay close attention to parentheses when they appear to ensure the calculations follow their intended sequence.