An algebraic expression uses letters, numbers, and operations to represent a mathematical relationship — but unlike an equation, it has no equals sign. Writing expressions from words is the bridge from arithmetic to algebra: you replace unknown quantities with letters and translate everyday language into mathematical notation.

What is an algebraic expression?

An expression is a collection of terms — each term is a number, a variable (letter), or a product of both. Terms are separated by + or − signs.

Examples of expressions: 3x, 2y + 5, 4a − b + 7, x²+ 3x − 2

Not expressions (these are equations): 3x = 12, y + 5 = 2y − 1

In algebra, letters stand for unknown or variable quantities. The letter n often represents a number; x and y are common for unknowns; letters like l, w, and h are used for lengths (length, width, height) and t for time.

How do you translate words into algebra?

Certain English phrases translate directly into algebraic operations:

Word phrase Algebraic meaning Example
"5 more than n" n + 5 "5 more than a number" → n + 5
"3 less than x" x − 3 "3 less than a price" → x − 3
"double y" / "twice y" 2y "double the cost" → 2c
"a third of m" m ÷ 3 or m/3 "a third of the distance" → d/3
"the product of p and q" pq —
"7 more than double n" 2n + 7 —
"the square of x" x² —

Key convention: write 3n rather than 3 × n or n3. The number (coefficient) always comes before the letter.

Worked example 1: expressions from word descriptions

Write an algebraic expression for each of the following:

  1. "Five more than a number n" → n + 5
  2. "Three times a number x, minus 8" → 3x − 8
  3. "The sum of two consecutive integers, where the first is n" → n and n + 1, so sum = n + (n + 1) = 2n + 1
  4. "A number y is halved, then 4 is added" → y/2 + 4 (or ½y + 4)

How do you write expressions for perimeter and area?

Geometric problems are a classic source of algebraic expressions at KS3.

Perimeter of a rectangle: If the length is l cm and the width is w cm:

  • Perimeter = l + w + l + w = 2l + 2w or equivalently 2(l + w)

Example: A rectangle has length (2n + 1) cm and width (n − 3) cm. Write an expression for the perimeter.

  • Perimeter = 2(2n + 1) + 2(n − 3)
  • = 4n + 2 + 2n − 6
  • = 6n − 4 cm ✓

Perimeter of a triangle: with sides a, b, and c: Perimeter = a + b + c. If two sides of an isosceles triangle are 3x cm and the base is (x + 4) cm:

  • Perimeter = 3x + 3x + (x + 4) = 7x + 4 cm ✓

Worked example 2: expressions from a real-life context

A cinema charges £8 for an adult ticket and £5 for a child ticket.

  1. Write an expression for the cost of a adult tickets and c child tickets. → 8a + 5c (in £)

  2. Three friends each buy a coffee for £p and a cake for £(p − 1). Write an expression for the total cost. → Each friend pays p + (p − 1) = 2p − 1; total for three friends = 3(2p − 1) = 6p − 3 (in £)

  3. A plumber charges a £35 call-out fee plus £h per hour. Write an expression for the cost of a 4-hour job. → 4h + 35 (in £)

What notation rules should you follow?

  • Write 5n, not 5 × n or n × 5.
  • Write n/3 or ⅓n, not n ÷ 3 in a final answer.
  • Write n², not n^2 in handwriting (use the superscript).
  • Do not write 1n — just write n.
  • Do not write −1n — write −n.
  • Collect like terms in your final expression: 3n + 2 + n − 5 = 4n − 3.

What mistakes do students make?

  • Reversing subtraction. "3 less than n" is n − 3, NOT 3 − n. The quantity being reduced comes first.
  • Forgetting to collect like terms. An expression like n + n + 2 should be simplified to 2n + 2.
  • Using × between a number and a letter. Write 5x not 5 × x.
  • Treating the expression as an equation. Do not solve or add "= 0" unless the question explicitly forms an equation.

Frequently asked questions

What is the difference between an expression and a formula?

An expression is a mathematical phrase with no equals sign: e.g. 3n + 2. A formula uses an equals sign to define one variable in terms of others: e.g. P = 2l + 2w. A formula tells you how to calculate something; an expression simply represents a quantity. Both use the same algebraic notation.

What is the difference between a term and a coefficient?

A term is a single piece of an expression, separated from others by + or −. In 5x² − 3x + 7, the three terms are 5x², −3x, and 7. The coefficient is the numerical part of a term: in 5x², the coefficient is 5; in −3x, the coefficient is −3. The constant term 7 has no variable part.

How do you write an expression for consecutive even numbers?

If n is any even number, the next even number is n + 2, then n + 4, and so on. Three consecutive even numbers starting at n can be written as n, n + 2, and n + 4. Their sum is 3n + 6 = 3(n + 2).

Can you evaluate an expression?

Yes — substituting a specific value for the variable gives a numerical result. If n = 4, then 3n − 5 = 3(4) − 5 = 12 − 5 = 7. This is called substitution and bridges expressions back to arithmetic.

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