Beyond the whole numbers you first learned to count with, there's a wider world of numbers called integers. Integers include all the whole numbers, their negative counterparts, and zero. Think of a number line: zero is in the middle, positive numbers stretch to the right, and negative numbers stretch to the left.

Adding and subtracting integers is like taking steps on this line. Adding a positive number means moving to the right. Adding a negative number (or subtracting a positive one) means moving to the left. For example, if you're at -3 and you add 5, you move five steps to the right, landing on 2.

Multiplication and division have their own simple rules.

The same rules apply to division. A positive divided by a positive is positive, and so on. The key thing to remember is that two negatives make a positive when multiplying or dividing.

Then we have rational numbers. These are any numbers that can be written as a fraction, like $1/2$, or as a decimal that either ends or repeats, like $0.5$ or $0.333...$ All integers are also rational numbers because you can write any integer as a fraction with a denominator of 1 (for example, $5 = 5/1$).

When adding or subtracting fractions, you need a common denominator. This means the bottom number of both fractions must be the same before you can combine the top numbers (numerators).

Multiplying fractions is more direct. You just multiply the numerators together and the denominators together. To divide by a fraction, you flip the second fraction upside down (find its reciprocal) and then multiply.

A percentage is just a special kind of fraction where the denominator is always 100. The word "percent" literally means "per hundred." So, 50% is the same as the fraction 50/100, which simplifies to 1/2. As a decimal, it's 0.50.

Understanding how to convert between percentages, fractions, and decimals makes them much easier to work with. To change a percent to a decimal, you just move the decimal point two places to the left. To change a decimal to a percent, you move it two places to the right.

Percentages are useful for figuring out discounts, tips, taxes, and much more. Imagine you want to buy a shirt that costs $40, but it's on sale for 25% off. How much do you save?

First, convert the percentage to a decimal: $25\% = 0.25$. Then, multiply the original price by the decimal: $\text{\textdollar}40 \times 0.25 = \text{\textdollar}10$.

You save $10 on the shirt. The sale price is $\text{\textdollar}40 - \text{\textdollar}10 = \text{\textdollar}30$.

Algebra introduces the idea of using symbols, usually letters, to represent unknown values. This lets us describe relationships and solve problems in a more general way. We call these symbols variables.

When we combine variables with numbers and operations like addition or subtraction, we create an algebraic expression. For example, $x + 5$ is an expression. It means "five more than some number x."

An equation is different. It sets two expressions equal to each other with an equals sign. For example, $x + 5 = 12$ is an equation. Our goal is usually to solve the equation, which means finding the value of the variable that makes the statement true.

To solve an equation, you need to keep it balanced. Whatever you do to one side of the equals sign, you must also do to the other side. To find the value of $x$ in $x + 5 = 12$, you need to get $x$ by itself. Since 5 is being added to $x$, you do the opposite operation: subtract 5 from both sides.

The same principle applies to other operations. If an equation involves multiplication, you use division to solve it. For the equation $3y = 21$, you would divide both sides by 3 to find that $y = 7$.

Ready to test your understanding? Let's try a few questions.

Mastering these fundamentals of integers, percentages, and basic algebra provides a solid foundation for tackling more complex mathematical challenges.