Math Word Problem Keywords: The Ultimate Cheat Sheet
If word problems make you freeze, it’s usually not the math — it’s the translation. For many students, translating English into math is the hardest part of algebra. You can know how to solve equations and still get stuck because the sentence feels “wordy.”
This guide is a practical, classroom-tested list of math word problem keywords that often signal specific operations. Use it as an algebra keywords cheat sheet when translating algebraic expressions into equations you can actually solve. And remember: keywords are clues, not magic spells — context matters.
The Big Four Operations Keyword Tables
Most word problems boil down to four operations: addition, subtraction, multiplication, and division. Each section below shows common keywords, a sample phrase, and the matching math equation. Use these tables like a quick “decoder” for your word problem solver workflow.
Addition Keywords ($+$)
Addition usually combines quantities or increases a value.
| Keyword | Example Phrase → Math Equation |
|---|---|
| Sum | The sum of $x$ and $8$ → $x + 8$ |
| Increased by | A number increased by $12$ → $n + 12$ |
| Total | The total of $15$ and $y$ → $15 + y$ |
| Combined | Combined length of $a$ and $b$ → $a + b$ |
| Plus | $k$ plus $3$ → $k + 3$ |
| Added to | $7$ added to $m$ → $7 + m$ |
Subtraction Keywords ($-$)
Subtraction often compares two amounts, removes a quantity, or decreases a value.
| Keyword | Example Phrase → Math Equation |
|---|---|
| Difference | The difference between $x$ and $9$ → $x – 9$ |
| Decreased by | A number decreased by $5$ → $n – 5$ |
| Minus | $t$ minus $4$ → $t – 4$ |
| Less than | $6$ less than $y$ → $y – 6$ |
| Take away | Take away $3$ from $p$ → $p – 3$ |
Multiplication Keywords ($\times$)
Multiplication shows repeated groups, scaling, area, and many percent situations. One keyword to treat carefully is “of,” which frequently means multiplication in algebra and percents.
| Keyword | Example Phrase → Math Equation |
|---|---|
| Product | The product of $7$ and $x$ → $7x$ |
| Times | $3$ times a number → $3n$ |
| Of | $30\%$ of $50$ → $0.30 \times 50$ |
| Twice | Twice the number $m$ → $2m$ |
| Double | Double $x$, then add $1$ → $2x + 1$ |
Division Keywords ($\div$)
Division usually means splitting into equal groups, rates, or ratios.
| Keyword | Example Phrase → Math Equation |
|---|---|
| Quotient | The quotient of $x$ and $5$ → $\frac{x}{5}$ |
| Ratio | The ratio of $a$ to $b$ → $\frac{a}{b}$ |
| Per | $60$ miles per $2$ hours → $\frac{60}{2}$ |
| Split equally | Split $24$ candies equally among $n$ kids → $\frac{24}{n}$ |
| Out of | $3$ out of $12$ → $\frac{3}{12}$ |
Equality Keywords ($=$)
Many students can translate operations but forget to spot the “equals” signal. These words often mean you should write an equals sign.
- Is
- Yields
- Results in
- Amounts to
- Equals
- Is the same as
Example: “The sum of $x$ and $4$ is $10$” → $x + 4 = 10$.
Warning: The “Trap” Words That Flip the Order
Some subtraction phrases are turnaround words. They tell you the order is reversed compared to how the sentence is spoken. This is a top reason students miss points when translating algebraic expressions.
Two common traps:
- Less than
- Subtracted from
Example: “5 less than $x$” means start with $x$, then subtract $5$ → $x – 5$. It does not mean $5 – x$.
Another example: “12 subtracted from $y$” means $y – 12$. The phrase “subtracted from” flips the order.
Quick check: If you can rewrite the phrase as “start with ___, then subtract ___,” you’ll usually get the order right.
Real-World Practice: Translate 3 Complex Sentences Step-by-Step
Let’s apply the cheat sheet to realistic sentences. Each example shows a clean translation path so you can build the equation confidently. If you’re using a word problem solver, these steps also help you verify the tool’s output and understand the math.
Practice 1: Multiple Operations with a Turnaround Phrase
Sentence: “Eight less than three times a number is twenty.”
- Choose a variable: let the number be $n$.
- Translate “three times a number”: → $3n$
- Translate “eight less than 3n”: → $3n – 8$ (turnaround phrase)
- Translate “is twenty”: → $= 20$
Final equation:
Practice 2: Ratios, Totals, and a Real-World Context
Sentence: “The ratio of x to 4, increased by 6, amounts to 15.”
- Translate “the ratio of x to 4”: → $\frac{x}{4}$
- Translate “increased by 6”: → $\frac{x}{4} + 6$
- Translate “amounts to 15”: → $= 15$
Final equation:
Practice 3: Percents, “Of,” and a Two-Step Relationship
Sentence: “Thirty percent of a number, decreased by 9, yields the same result as twice the number plus 3.”
- Let the number be: $n$
- Translate “thirty percent of a number”: → $0.30n$ (because “of” means multiplication)
- Translate “decreased by 9”: → $0.30n – 9$
- Translate “twice the number plus 3”: → $2n + 3$
- Translate “yields the same result as”: → $=$
Final equation:
Conclusion
Word problems get easier when you stop trying to “feel” your way through them and start translating them systematically. This algebra keywords cheat sheet gives you the most common operation signals, the equality keywords that trigger an equals sign, and the trap words that flip subtraction order.
With practice, translating algebraic expressions becomes a skill you can trust — not a guess.
Frequently Asked Questions
A: Turnaround words are subtraction phrases like “less than” and “subtracted from” that reverse the order of the terms. For example, “5 less than x” translates to $x – 5$, not $5 – x$.
A: In algebra and percentage problems, the word “of” almost always indicates multiplication. For example, “30% of 50” translates to $0.30 \times 50$.
A: Words like “is”, “yields”, “results in”, “amounts to”, and “is the same as” signal that you need to write an equals sign ($=$) in your algebraic equation.
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