Word Problems: Writing Algebraic Expressions
Master the art of translating English into the language of algebra — the single most important skill for conquering SAT word problems.
1 What Is an Algebraic Expression?
✅ Algebraic Expression
A mathematical phrase containing numbers, variables, and operations that represents a quantity.
- • $3x + 5$
- • $\frac{n}{4} - 7$
- • $2(a + b)$
No equals sign. It's a phrase, not a sentence.
⚠️ Equation (Different!)
An equation states that two expressions are equal. It contains an equals sign.
- • $3x + 5 = 20$
- • $\frac{n}{4} - 7 = 1$
- • $2(a + b) = 14$
Has equals sign. It's a complete sentence.
🔤 Variables Represent Unknowns
A variable is a letter that stands for an unknown quantity. Common choices: $x$, $y$, $n$. SAT Tip: Choose meaningful letters when helpful — $a$ for age, $d$ for distance, $t$ for time, $c$ for cost.
2 The Complete Translation Guide: Words → Math
Memorize these keyword-to-operation translations. They appear on every single SAT.
➕ Addition Keywords
Keywords: sum plus more than increased by added to total combined together exceeds by
| English | Algebra |
|---|---|
| "5 more than $x$" | $x + 5$ |
| "sum of $a$ and $b$" | $a + b$ |
| "$n$ increased by 7" | $n + 7$ |
➖ Subtraction Keywords
Keywords: difference minus less than decreased by subtracted from fewer than reduced by take away
| English | Algebra |
|---|---|
| "$x$ decreased by 4" | $x - 4$ |
| "7 less than $n$" | $n - 7$ |
| "difference of $a$ and $b$" | $a - b$ |
"5 less than $x$" means $x - 5$ (start with $x$, subtract 5)
"$x$ less than 5" means $5 - x$ (start with 5, subtract $x$)
The number after "less than" comes first in the subtraction. This is the #1 SAT trap!
✖️ Multiplication Keywords
Keywords: product times multiplied by of twice double triple
| English | Algebra |
|---|---|
| "product of 3 and $x$" | $3x$ |
| "twice $n$" | $2n$ |
| "half of $x$" | $\frac{x}{2}$ |
➗ Division Keywords
Keywords: quotient divided by ratio per split among shared equally out of
| English | Algebra |
|---|---|
| "quotient of $x$ and 5" | $\frac{x}{5}$ |
| "$n$ divided by 3" | $\frac{n}{3}$ |
| "ratio of $a$ to $b$" | $\frac{a}{b}$ |