How do you translate English phrases into algebraic expressions?

To write an algebraic expression, you translate an English sentence word by word into mathematical symbols. Addition words include sum, plus, and increased by. Subtraction words include difference, minus, and decreased by. Multiplication words include product, times, and of. Division words include quotient, divided by, and per.

Does word order matter in algebraic expressions?

Yes, word order matters for subtraction and division. 'b decreased by 15' means b minus 15, not 15 minus b. '5 less than x' means x minus 5, not 5 minus x. 'The quotient of p and q' means p divided by q, not q divided by p.

Writing algebraic expressions

Q: What is an algebraic expression?

An algebraic expression uses letters (called variables) together with numbers and operation symbols. Variables such as a, b, x or n stand for numbers whose value we do not yet know, or which can change.

Learn how to translate everyday English phrases into algebraic expressions. This page explains the key vocabulary, the conventions used in algebra, and walks through four worked examples to build your confidence.

What is an Algebraic Expression?

An algebraic expression uses letters (called variables) together with numbers and operation symbols. Variables such as $a$ , $b$ , $x$ or $n$ stand for numbers whose value we do not yet know, or which can change.

To write an algebraic expression, you translate an English sentence, word by word, into mathematical symbols. The key is to recognize which operation each word is describing.

Translating words to operations

Different English words map to the four basic operations:

Addition ( $+$ ): sum, plus, add, increased by, more than, total
Subtraction ( $-$ ): difference, minus, less than, decreased by, subtract, take away
Multiplication ( $\cdot$ ): product, times, multiplied by, of, twice, triple
Division ( $\div$ or a fraction): quotient, divided by, per, ratio, one half of, one third of

Powers and Roots

"Squared" means raised to the power of $2$ , e.g. $a$ squared is $a^{2}$ .
"Cubed" means raised to the power of $3$ , e.g. $a$ cubed is $a^{3}$ .
"The number multiplied by itself $n$ times" is $a^{n}$ . For example, $a \cdot a \cdot a \cdot a = a^{4}$ .
"The square root of" a quantity is written with a radical sign, e.g. the square root of $x$ is $\sqrt{x}$ .

Shorthand conventions in algebra

When writing algebraic expressions, we use these standard shortcuts:

The multiplication sign is usually omitted. Write $3 m$ instead of $3 \cdot m$ , and $a b$ instead of $a \cdot b$ .
The number is written first, then the letter: write $5 p$ , not $p 5$ .
Division is written as a fraction: write $\frac{p}{q}$ rather than $p \div q$ .
For groups, use parentheses to show which terms are combined first. For example, "one third of the sum of $a$ and $b$ " is $\frac{a + b}{3}$ , not $a + \frac{b}{3}$ .

⚠️ Watch out: word order matters

Be careful with subtraction and division — the order matters.

" $b$ decreased by $15$ " means $b - 15$ , not $15 - b$ .
" $5$ less than $x$ " means $x - 5$ , not $5 - x$ .
"The quotient of $p$ and $q$ " means $\frac{p}{q}$ , not $\frac{q}{p}$ .

Worked examples

Example 1 — Translating with a Power

If $x$ represents a number, write an algebraic expression for the number multiplied by itself $4$ times, then decreased by nine.

Solution

Translate the sentence piece by piece:

\begin{aligned} Multiplied by itself 4 times & = x \cdot x \cdot x \cdot x \\ = x^{4} \\ Decreased by nine & \Rightarrow - 9 \\ ∴ expression & = x^{4} - 9 \end{aligned}

Example 2 — One Quarter of a Sum

Write an expression for one quarter of the sum of $m$ and $n$ .

Solution

First form the sum, then divide by $4$ . Brackets (or a fraction bar) are needed so the whole sum is divided:

\begin{aligned} The sum of m and n & = m + n \\ One quarter of the sum & = \frac{m + n}{4} \end{aligned}

Example 3 — A Geometric Formula

Write an expression for the perimeter of an equilateral triangle with side length $s$ .

Solution

An equilateral triangle has three equal sides, so the perimeter is three lots of $s$ :

\begin{aligned} P & = s + s + s \\ = 3 s \end{aligned}

Example 4 — A Money Word Problem

Liam buys $4$ notebooks costing $$ b$ each and pays with a $$ k$ note. Write an expression for the change he receives.

Solution

Find the total cost first, then subtract it from the amount paid:

\begin{aligned} Cost of 1 notebook & = $ b \\ Cost of 4 notebooks & = $ 4 b \\ Amount paid & = $ k \\ Change & = $ (k - 4 b) \end{aligned}

Key rules to remember

Match the word to the operation: sum → $+$ , difference → $-$ , product → $\cdot$ , quotient → $\div$ .
Drop the multiplication sign. Write $3 m$ instead of $3 \cdot m$ , and $x y$ instead of $x \cdot y$ .
Number first, letter second: write $5 p$ , not $p 5$ .
Word order matters for $-$ and $\div$ : " $a$ decreased by $b$ " is $a - b$ , not $b - a$ . "The quotient of $p$ and $q$ " is $\frac{p}{q}$ , not $\frac{q}{p}$ .
Write division as a fraction using $\frac{}{}$ rather than $\div$ .
Use parentheses (or a fraction bar) to group terms that belong together. "Half the sum of $a$ and $b$ " is $\frac{a + b}{2}$ , not $\frac{a}{2} + b$ .
Powers mean repeated multiplication: $a \cdot a \cdot a \cdot a \cdot a = a^{5}$ . "Squared" means $^{2}$ , "cubed" means $^{3}$ .
For word problems, identify the cost/quantity of one item first, then scale up and combine.

Writing algebraic expressions

📖 Theory

What is an Algebraic Expression?

Powers and Roots

Example 1 — Translating with a Power

Example 2 — One Quarter of a Sum

Example 3 — A Geometric Formula

Example 4 — A Money Word Problem

Theory