Introduction

In arithmetic, you work mostly with specific numbers like $3$, $12$, or $100$.
In algebra, we often don’t know a number yet — but we still want to talk about it, reason about it, or compute with it.
To do this, we use letters.

This article introduces:

• What variables are
• What constants are
• Why letters are useful
• How to read simple algebraic expressions
• How this prepares you for later articles on writing and evaluating expressions

Why Use Letters in Mathematics?

Letters allow us to:

• Represent numbers we don’t know yet
• Describe patterns and general rules
• Write formulas that work for many situations
• Avoid repeating long explanations

Examples:

• “A number increased by $5$” can be written as $x + 5$
• “Twice a number” becomes $2x$
• “The cost of $n$ tickets at £4 each” becomes $4n$

Letters make ideas shorter, clearer, and more flexible.

Variables

A variable is a letter that stands for a number that can change or is unknown.

Common variable letters:

• $x$
• $y$
• $n$
• $t$

Key ideas:

• A variable is a placeholder for a number.
• The value might be unknown, or it might vary from one situation to another.
• Variables let us write general statements like “the area of a square is $s^2$”.

Examples:

• If $x$ represents “a number”, then $x + 3$ means “that number plus $3$”.
• If $t$ represents time in seconds, then $5t$ means “five times the number of seconds”.

Constants

A constant is a number that does not change.

Examples:

• $2$, $10$, $-7$, $0.5$
• In the expression $3x + 4$, the number $4$ is a constant.
• In the formula for the area of a circle, $A = \pi r^2$, the number $\pi$ is a constant.

Key points:

• Constants stay the same.
• Variables may change.
• Expressions often mix both.

Combining Variables and Constants

When we combine variables and constants using arithmetic operations, we create algebraic expressions.

Examples:

• $x + 7$
• $2n - 3$
• $5t$
• $10 + k$

Important notes:

• Writing $5x$ means $5 \times x$.
• We usually omit the multiplication symbol when multiplying a number and a variable.
• Expressions are not equations — they don’t have an equals sign.

Simple Examples

Here are some everyday situations and how variables help describe them:

Exercises

1. Write “a number increased by $4$” using a variable.

   Solution

   “A number increased by $4$” → $x + 4$
   We use $x$ to represent the unknown number.
2. If $n$ is the number of books you buy, and each book costs £6, write an expression for the total cost.

   Solution

   Total cost for $n$ books at £6 each → $6n$
   Multiply price by quantity.
3. Write an expression for “twice a number minus $3$”.

   Solution

   “Twice a number minus $3$” → $2x - 3$
   Twice a number is $2x$.
4. If $t$ represents time in minutes, write an expression for “$5$ minutes more than $t$”.

   Solution

   “$5$ minutes more than $t$” → $t + 5$
   Add $5$ to the variable.
5. A rectangle has width $w$. Write an expression for its perimeter if the length is $10$.

   Solution

   Perimeter of rectangle with width $w$ and length $10$ → $2w + 20$
   Perimeter is $2(\text{width}) + 2(\text{length})$.
6. Write an expression for “half of a number $k$ plus $1$”.

   Solution

   “Half of $k$ plus $1$” → $\frac{k}{2} + 1$
   Half of $k$ is $\frac{k}{2}$.
7. If $m$ is the number of miles you walk, and each mile burns $80$ calories, write an expression for total calories burned.

   Solution

   Calories burned walking $m$ miles at $80$ calories per mile → $80m$
   Multiply calories per mile by miles walked.
8. Write “a number decreased by $9$” using a variable.

   Solution

   “A number decreased by $9$” → $x - 9$
   Subtract $9$ from the variable.