Writing Expressions from Words

Q: Does “the quotient of $y$ and $3$” mean $y \div 3$ or $3 \div y$?

It means $y \div 3$, written $\dfrac{y}{3}$. In “the quotient of $a$ and $b$,” the first quantity named is the one being divided — it goes on top of the fraction.

Q: How can I check that my expression is right?

Test it with a real number. For “$4$ less than twice $x$,” try $x = 10$: twice $10$ is $20$, and $4$ less than that is $16$. Your expression $2x - 4$ gives $2(10) - 4 = 16$ — it matches, so the translation is sound.

Word problems don't hand you an expression — they hand you a phrase like “ $7$ more than a number” and expect you to build the algebra yourself. Translating words into symbols is its own skill, and it runs on a short dictionary: sum means add, difference means subtract, product means multiply, quotient means divide.

Most of the dictionary is direct — you write things in the order you hear them. One phrase breaks the pattern: “less than.” It flips the order of the subtraction, and it is the trap this lesson exists to catch.

The key words

Sum means add: “the sum of $x$ and $9$ ” is $x + 9$ . Difference means subtract: “the difference of $x$ and $5$ ” is $x - 5$ . So does “decreased by”: “ $n$ decreased by $2$ ” is $n - 2$ .

Product means multiply: “the product of $5$ and $x$ ” is $5x$ , and “twice a number” means $2$ times it. Quotient means divide: “the quotient of $y$ and $3$ ” is $\dfrac{y}{3}$ — whatever is named first goes on top.

“Less than” flips the order

“ $3$ less than $x$ ” is $x - 3$ , not $3 - x$ . The phrase means you start with $x$ and take $3$ away from it, so the $3$ is written last — the opposite of the order you hear it. “Fewer than” and “subtracted from” flip the same way.

The flip still applies when the starting amount is bigger than a single letter. “ $6$ less than the product of $4$ and $x$ ” builds the product first, $4x$ , then takes $6$ away from it: $4x - 6$ .

Build in the order the phrase builds

Longer phrases stack operations, so find the main quantity first and then apply what is done to it. “ $4$ less than twice a number $x$ ” starts with twice the number, $2x$ , then subtracts: $2x - 4$ . That is not the same as $2(x - 4)$ — the doubling happened before the subtraction, so nothing gets grouped.

Parentheses appear only when a whole sum or difference gets multiplied or divided. “Twice the sum of $x$ and $3$ ” doubles the entire sum, so it is written $2(x + 3)$ . Compare it with “the sum of twice $x$ and $3$ ,” which is $2x + 3$ — the words tell you what was built first.

Worked examples

Example 1: more than

Write an expression for “ $7$ more than a number $n$ .”

Start with the number

n

“More than” means add, so add

7

to it

n + 7

Answer: $n + 7$

Example 2: a product

Write an expression for “the product of $5$ and a number $x$ .”

“Product” means multiply the two quantities named

5 \cdot x

Write the multiplication without the dot

5x

Answer: $5x$

Example 3: the flip

Write an expression for “ $4$ less than twice a number $x$ .”

“Twice a number

x

” means multiply by

2

2x

“

4

less than” takes

4

away from that amount, so the

4

goes last

2x - 4

Answer: $2x - 4$

Try one yourself

Common questions

Why does “less than” flip the order?

Because the phrase names the amount being taken away first. “ $3$ less than $x$ ” describes $x$ with $3$ removed, so the subtraction is written $x - 3$ . Subtraction order matters: $x - 3$ and $3 - x$ are different numbers.

When do I need parentheses?

When an entire sum or difference is multiplied or divided as one piece. “Twice the sum of $x$ and $3$ ” is $2(x + 3)$ because the whole sum gets doubled. If only the number is doubled, as in “the sum of twice $x$ and $3$ ,” no parentheses are needed: $2x + 3$ .

Does “the quotient of

y

and

3

” mean

y \div 3

3 \div y

It means $y \div 3$ , written $\dfrac{y}{3}$ . In “the quotient of $a$ and $b$ ,” the first quantity named is the one being divided — it goes on top of the fraction.

How can I check that my expression is right?

Test it with a real number. For “ $4$ less than twice $x$ ,” try $x = 10$ : twice $10$ is $20$ , and $4$ less than that is $16$ . Your expression $2x - 4$ gives $2(10) - 4 = 16$ — it matches, so the translation is sound.

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Writing Expressions from Words

The key words

“Less than” flips the order

Build in the order the phrase builds

Worked examples

Example 1: more than

Example 2: a product

Example 3: the flip

Try one yourself

Common questions

Keep learning

Variables & Algebraic Expressions

Evaluating Expressions

Combining Like Terms

Two-Step Equations