Basic Arithmetic & Algebra Examples

Work through these examples to deepen your understanding of basic arithmetic and algebra symbols.

Symbol Usage Examples

Here are common usage examples for each symbol:

Plus Sign ( $+$ ): Addition ( $3 + 2 = 5$ )
Minus Sign ( $-$ ): Subtraction ( $3 - 2 = 1$ ) or negative numbers ( $-2$ )
Times Sign ( $\times$ or $\cdot$ ): Multiplication ( $3 \times 2 = 6$ or $3 \cdot 2 = 6$ )
Division Sign ( $\div$ or $/$ ): Division ( $6 \div 2 = 3$ or $6/2 = 3$ )
Equals ( $=$ ): Equality; values on both sides are the same ( $2 + 2 = 4$ )
Not Equal ( $\neq$ ): The values are not the same ( $2 \neq 3$ )
Approximately Equal ( $\approx$ ): Close to, but not exactly ( $\pi \approx 3.14$ )
Plus-Minus ( $\pm$ ): Both positive and negative operations ( $x = \pm 5$ )
Square Root ( $\sqrt{x}$ ): A number that produces $x$ when multiplied by itself ( $\sqrt{4} = 2$ )
Absolute Value ( $|x|$ ): Distance from zero ( $|-3| = 3$ )
Factorial ( $n!$ ): Product of integer and all integers below it ( $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ )

Order of Operations (PEMDAS)

When an equation contains multiple symbols (like plus, times, and exponents), you cannot just calculate from left to right. You must follow a specific hierarchy to get the correct answer.

The Hierarchy:

Parentheses: Solve everything inside grouping symbols first.
Exponents: Calculate powers and square roots.
Multiplication & Division: These are tied in priority. Solve them from left to right.
Addition & Subtraction: These are also tied. Solve them from left to right.

Problem 1: Solve $2 + 3 \times (4 - 1)^2 - 5$

Solution:

Step 1 (Parentheses): $4 - 1 = 3$
Equation becomes: $2 + 3 \times 3^2 - 5$
Step 2 (Exponents): $3^2 = 9$
Equation becomes: $2 + 3 \times 9 - 5$
Step 3 (Multiplication): $3 \times 9 = 27$
Equation becomes: $2 + 27 - 5$
Step 4 (Addition & Subtraction): $2 + 27 - 5 = 24$

Answer: $24$

Problem 2: Evaluate $\frac{8 + 2 \times 3^2}{4 - 1}$

Solution:

Step 1: Calculate numerator: $8 + 2 \times 3^2 = 8 + 2 \times 9 = 8 + 18 = 26$
Step 2: Calculate denominator: $4 - 1 = 3$
Step 3: Divide: $\frac{26}{3}$

Answer: $\frac{26}{3}$ or approximately $8.67$

Problem 3: Simplify $|x - 5|$ when $x = 2$ and when $x = 8$

Solution:

When $x = 2$ : $|2 - 5| = |-3| = 3$
When $x = 8$ : $|8 - 5| = |3| = 3$

Answer: Both equal $3$

Daily Life Applications

Understanding basic arithmetic and algebra symbols helps you solve real-world problems every day. Here are practical ways to use these concepts:

Budgeting and Shopping

Addition ( $+$ ): Calculate total costs when shopping. If you buy groceries for $45, gas for $30, and coffee for $5, your total is $45 + 30 + 5 = 80$ dollars.
Subtraction ( $-$ ): Track remaining budget. If you have $200 and spend $80, you have $200 - 80 = 120$ dollars left.
Multiplication ( $\times$ ): Calculate costs for multiple items. If one item costs $12 and you need $5$ of them, the total is $12 \times 5 = 60$ dollars.
Division ( $\div$ ): Split bills evenly. If a dinner costs $120 and is split among $4$ friends, each pays $120 \div 4 = 30$ dollars.

Cooking and Recipes

Ratios and Proportions: Adjust recipe quantities. If a recipe serves $4$ people but you need to serve $6$ , multiply all ingredients by $\frac{6}{4} = 1.5$ .
Temperature Conversions: Convert between Celsius and Fahrenheit using formulas like $F = \frac{9}{5}C + 32$ .

Time Management

Addition: Calculate total travel time. If your commute is $25$ minutes to work and $30$ minutes back, total is $25 + 30 = 55$ minutes per day.
Subtraction: Determine how much time you have. If you have $2$ hours ( $120$ minutes) and need $45$ minutes for tasks, you have $120 - 45 = 75$ minutes free.

Distance and Travel

Multiplication: Calculate total distance. If you drive $15$ miles to work each way, $5$ days a week, you drive $15 \times 2 \times 5 = 150$ miles per week.
Division: Calculate average speed. If you travel $60$ miles in $1.5$ hours, your average speed is $\frac{60}{1.5} = 40$ miles per hour.

Health and Fitness

Percentages: Track nutrition goals. If you need $2000$ calories and have consumed $1500$ , you have $\frac{1500}{2000} = 0.75 = 75\%$ of your daily goal.
Absolute Value ( $|x|$ ): Measure differences. If your target weight is $150$ lbs and you weigh $148$ lbs, you're $|148 - 150| = 2$ lbs away from your goal.

Home Improvement

Area Calculations: Calculate paint needed. If a room is $12$ feet by $10$ feet, the area is $12 \times 10 = 120$ square feet.
Perimeter: Determine material needed for borders. A rectangular garden $8$ feet by $6$ feet has a perimeter of $2(8 + 6) = 28$ feet.

Problem-Solving Strategy

When facing daily problems:

Identify what you're solving for (total cost, time needed, etc.)
List the information you have (prices, quantities, rates)
Choose the right operation (addition for totals, multiplication for repeated amounts)
Check your answer by working backwards or estimating

These symbols aren't just abstract concepts—they're tools for making better decisions in everyday life!