Translating Words to Algebraic Expressions

One of the biggest hurdles in algebra is not the math itself — it is turning an English sentence into a mathematical expression. Once you can do that, the simplification and solving steps follow the rules you already know. This page teaches you to recognize the key phrases that signal each operation and translate them accurately.

Key Phrases for Each Operation

Addition Phrases

Subtraction Phrases

Watch the order with subtraction! “Four less than a number” means $n - 4$, NOT $4 - n$. The phrase “less than” reverses the written order — the number after “than” comes first in the expression.

Multiplication Phrases

Division Phrases

Step-by-Step Translation

Example 1: “Three more than twice a number”

Break the sentence into pieces:

1. “A number” — let it be $n$
2. “Twice a number” — $2n$
3. “Three more than” — add 3

$2n + 3$

Answer: $2n + 3$

Example 2: “The quotient of a number and six, decreased by four”

1. “A number” — $n$
2. “The quotient of a number and six” — $\frac{n}{6}$
3. “Decreased by four” — subtract 4

$\frac{n}{6} - 4$

Answer: $\dfrac{n}{6} - 4$

Example 3: “Five less than the product of eight and a number”

1. “A number” — $n$
2. “The product of eight and a number” — $8n$
3. “Five less than” — subtract 5 from the product

$8n - 5$

Answer: $8n - 5$

Remember: “less than” means subtraction in reverse order. The 5 is subtracted FROM $8n$.

Example 4: “The sum of a number and its square”

1. “A number” — $n$
2. “Its square” — $n^2$
3. “The sum of” — add them

$n + n^2$

Answer: $n + n^2$ (or equivalently, $n^2 + n$)

Translating Complete Sentences

When a sentence says “is” or “equals,” it becomes an equation (with an $=$ sign). For now, we are focusing on translating just the expression parts. But noticing where “is” appears tells you where the equals sign will go when you are ready to solve equations.

Example 5: “Seven more than three times a number is twenty-two”

The expression on the left side of “is”:

$3n + 7$

The full equation would be $3n + 7 = 22$, but we focus on building the expression correctly.

Common Patterns

Consecutive Integers

Three consecutive integers starting at $n$:

$n, \quad n + 1, \quad n + 2$

Three consecutive even integers starting at $n$ (where $n$ is even):

$n, \quad n + 2, \quad n + 4$

Three consecutive odd integers starting at $n$ (where $n$ is odd):

$n, \quad n + 2, \quad n + 4$

Notice consecutive even and consecutive odd integers use the same pattern — both skip by 2.

Age Problems Setup

If Maria is $x$ years old now:

• Her age 5 years ago: $x - 5$
• Her age in 10 years: $x + 10$
• Her brother who is twice her age: $2x$
• Her mother who is 25 years older: $x + 25$

Real-World Application: Nursing — Dosage Word Problem

A nurse reads: “The patient requires a dose that is 1.5 times their weight in kilograms, minus 20 milligrams for the age adjustment.”

Let $w$ represent the patient’s weight in kilograms.

1. “1.5 times their weight in kilograms” — $1.5w$
2. “Minus 20 milligrams” — subtract 20

$\text{Dose} = 1.5w - 20$

If the patient weighs $w = 80$ kg:

$\text{Dose} = 1.5(80) - 20 = 120 - 20 = 100 \text{ mg}$

Translating the written order into an algebraic expression ensures the nurse can calculate the correct dosage for any patient weight.

Real-World Application: Retail — Revenue Expression

A store owner says: “Our daily revenue is the number of customers times the average purchase, plus a flat $50 from online orders.”

Let $c$ represent the number of customers and $p$ represent the average purchase amount.

$\text{Revenue} = cp + 50$

On a day with 120 customers and an average purchase of $35:

$\text{Revenue} = 120(35) + 50 = 4{,}200 + 50 = 4{,}250$

The daily revenue is $4,250.

”More Than” vs. “Less Than” — Getting the Order Right

The phrases “more than” and “less than” are the most common sources of translation errors.

“More than” — add in either order (addition is commutative):

• “8 more than $n$” = $n + 8$
• This can also be written $8 + n$ — same result

“Less than” — order matters (subtraction is NOT commutative):

• “8 less than $n$” = $n - 8$ (start with $n$, subtract 8)
• This is NOT $8 - n$

Test yourself: “12 less than a number” means $n - 12$. Read it as: “Start with the number, then go 12 less.”

Common Mistakes to Avoid

1. Reversing “less than” order. “Six less than a number” is $n - 6$, not $6 - n$. The value after “than” comes first.

2. Ignoring grouping. “Twice the sum of a number and 3” is $2(n + 3)$, not $2n + 3$. The word “sum” tells you to add first, which requires parentheses.

3. Confusing “of” with addition. “One-third of a number” means $\frac{1}{3}n$ (multiplication), not $\frac{1}{3} + n$.

4. Translating “is” as part of the expression. The word “is” translates to $=$. It separates the two sides of an equation, not part of a single expression.

5. Forgetting that “a number” means a variable. Every time you see “a number” or “an unknown quantity,” assign a variable like $n$ or $x$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Each arithmetic operation has specific English phrases that signal it — learn the tables above
• “Less than” and “subtracted from” reverse the order — the number after “than” or “from” comes first in the expression
• Always assign a variable to “a number” or “an unknown” before translating
• Words like “sum” and “product” often imply parentheses when combined with other operations
• The word “is” translates to an equals sign ($=$), separating an equation into two sides
• Practice reading the phrase backwards to get subtraction order right: “5 less than $n$” means start with $n$, subtract 5
• In nursing, retail, and trades, translating written instructions into algebraic expressions is the first step to solving real problems

Return to Pre-Algebra for more topics in this section.