Arithmetic is the foundation of math. It all starts with four key operations: addition, subtraction, multiplication, and division. Mastering these is the first step to tackling more complex problems.

Let's quickly review what they do.

Addition (+) combines quantities. If you have 3 apples and get 2 more, you add them together to find the total: $3 + 2 = 5$ apples.

Subtraction (-) finds the difference between quantities or what's left after taking some away. If you start with 5 apples and eat 2, you have $5 - 2 = 3$ apples left.

Multiplication (×) is a shortcut for repeated addition. Instead of adding $4 + 4 + 4$, you can multiply $3 \times 4$ to get 12. It's a faster way to handle equal-sized groups.

Division (÷) is about splitting a quantity into equal parts. If you have 12 apples and want to share them among 3 friends, each person gets $12 \div 3 = 4$ apples.

Numbers follow certain rules, or properties, that make calculations predictable and often easier. These properties govern how the four operations relate to each other.

For example, $5 + 10$ is the same as $10 + 5$, and $7 \times 2$ is the same as $2 \times 7$. This property doesn't apply to subtraction or division, where order is very important.

If you're adding $2 + 3 + 4$, you can do $(2 + 3) + 4$ to get $5 + 4 = 9$, or you can do $2 + (3 + 4)$ to get $2 + 7 = 9$. The answer is the same.

This property is useful for breaking down problems. To calculate $5 \times 12$, you could think of 12 as $(10 + 2)$. Then, you can distribute the 5: $(5 \times 10) + (5 \times 2)$, which is $50 + 10 = 60$.

When a calculation involves multiple operations, you need to perform them in a specific sequence to get the right answer. We use the acronym PEMDAS to remember this order.

Let's solve an example: $3 + 5 \times (7 - 4)$.

1. Parentheses: First, we solve $(7 - 4)$, which equals 3.
2. The problem is now $3 + 5 \times 3$.
3. Multiplication: Next, we multiply $5 \times 3$, which equals 15.
4. The problem is now $3 + 15$.
5. Addition: Finally, we add $3 + 15$ to get 18.

Sometimes we need to work with parts of whole numbers. We can express these parts as fractions, decimals, or percentages. They are different ways of representing the same value.

Fractions, like $\frac{3}{4}$, show a relationship between a part (the numerator, 3) and a whole (the denominator, 4). This fraction means we have 3 out of 4 equal parts.

Decimals are another way to write fractions, but specifically those with denominators that are powers of 10 (like 10, 100, 1000). The fraction $\frac{3}{4}$ is written as 0.75 in decimal form.

Percentages represent a fraction out of 100. The symbol % means "per hundred." So, 75% is the same as the fraction $\frac{75}{100}$, which simplifies to $\frac{3}{4}$.

Being able to convert between these forms is a key skill. For example, knowing that $\frac{1}{2}$, 0.5, and 50% are all the same value allows you to choose the easiest form for solving a problem.

Let's practice what we've learned.

Understanding these arithmetic fundamentals is the key to unlocking more advanced math. They are the tools you'll use time and time again.