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\( x + 5 \)

Word Problems to
Algebraic Expressions

Learn to translate everyday language into the precise language of algebra: turning words into variables, operations, and expressions.

SSAT Middle Level: Quantitative

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables (like \( x \) or \( n \)), and operations (like \( + \), \( - \), \( \times \), \( \div \)) that represents a mathematical relationship. Unlike an equation, an expression has no equals sign.

Variable
A letter that stands for an unknown number, like \( x \) or \( n \).
Operation
An action performed on numbers: \( +, -, \times, \div \).
Constant
A fixed number that does not change, like \( 5 \) or \( 12 \).

Why Translate Words into Math?

Real-world problems come in words, but solving them requires math. Translation is the bridge: once you write an expression, you can evaluate it, simplify it, or use it in an equation to find an answer.

Word Problem Translate Algebraic Expression

The Key Vocabulary Dictionary

Certain English words always map to specific math operations. Memorize these signal words and you will translate faster.

Addition (\( + \))
sum, plus, more than, increased by, added to, total
Subtraction (\( - \))
difference, minus, less than, decreased by, fewer than
Multiplication (\( \times \))
product, times, of, twice, double, triple, each
Division (\( \div \))
quotient, divided by, ratio, per, split equally, half of

The Translation Rule

"Five more than a number" Unknown
a number \(\rightarrow\) \(n\)
Operation
more than \(\rightarrow\) \(+\)
Constant
five \(\rightarrow\) \(5\)
\( n + 5 \)

Order Matters: "Less Than" Trap

The phrase "less than" reverses the order of the numbers. Compare these two phrases carefully:

"5 less than \( x \)"
Means: start at \( x \) and subtract 5.
Expression: \( x - 5 \)
"5 minus \( x \)"
Means: start at 5 and subtract \( x \).
Expression: \( 5 - x \)

Expression vs. Equation

Students often confuse these two. An expression is a phrase; an equation is a complete sentence with an equals sign.

Expression Equation
Equals sign? No Yes
Example \( 3x + 7 \) \( 3x + 7 = 22 \)
Analogy A phrase: "three times a number plus seven" A sentence: "three times a number plus seven is twenty-two"
Can you solve? Evaluate or simplify only Solve for the unknown

The 4-Step Translation Method

1
Read the problem carefully. Underline key numbers and relationships.
2
Assign a variable to the unknown quantity: let \( n \) = the number, \( a \) = age, etc.
3
Identify the operation from signal words: "more than" → \( + \), "product" → \( \times \), etc.
4
Assemble the expression with correct order and grouping. Use parentheses when needed.

Example 1: Simple Addition

1
Unknown: "a number" → let \( n \) be the number.
2
Operation: "increased by" → addition (\( + \)).
3
Constant: 12.
4
Expression: \( n + 12 \).
\( n \)
\( + \)
\( 12 \)
\( = \)
\( n + 12 \)

Example 2: Multiplication + Subtraction

1
Unknown: "a number" → let \( n \) be the number.
2
Operation 1: "twice" → multiply by 2, giving \( 2n \).
3
Operation 2: "three less than" → subtract 3 from the result. Remember: "less than" reverses the order!
4
Expression: \( 2n - 3 \).

Example 3: Multi-Step with Parentheses

1
Inner operation first: "the sum of a number and 9" → \( n + 9 \).
2
Outer operation: "four times" that sum → multiply the entire sum by 4.
3
Parentheses: We need parentheses to show that 4 multiplies the whole sum, not just \( n \).
4
Expression: \( 4(n + 9) \).

Example 4: Real-World Scenario

1
Unknown: "some stickers" → let \( s \) = Mia's stickers.
2
Step 1: "gives away 8" → subtract 8: \( s - 8 \).
3
Step 2: "splits equally among 3" → divide by 3.
4
Expression: \( \dfrac{s - 8}{3} \). Parentheses (or a fraction bar) group the subtraction before dividing.

Common Mistakes to Avoid

✕ Reversing "Less Than"
Writing \( 7 - n \) for "7 less than \( n \)." Correct: \( n - 7 \). The number after "less than" comes first.
✕ Forgetting Parentheses
Writing \( 2n + 5 \) for "twice the sum of \( n \) and 5." Correct: \( 2(n + 5) \). Group the sum first!
✕ Wrong Operation
Confusing "more than" (addition) with "times as many" (multiplication). Read the signal word twice.

Translation Reference Table

English Phrase Operation Expression
a number increased by 6 \( + \) \( n + 6 \)
10 less than a number \( - \) \( n - 10 \)
the product of 5 and a number \( \times \) \( 5n \)
a number divided by 4 \( \div \) \( \dfrac{n}{4} \)
twice the sum of a number and 3 \( \times \) and \( + \) \( 2(n + 3) \)
half of a number decreased by 7 \( \div \) and \( - \) \( \dfrac{n}{2} - 7 \)

Quick Recap Before Practice

1
Variables represent unknowns: "a number" → \( n \).
2
Signal words tell you the operation: "sum" → \( + \), "product" → \( \times \), "less than" → \( - \), "divided by" → \( \div \).
3
"Less than" and "fewer than" reverse the order: "5 less than \( n \)" → \( n - 5 \).
4
Parentheses group operations that must happen first: "twice the sum" → \( 2(n + \ldots) \).
5
Check your work by substituting a number and seeing if the expression matches the story.

Question 1 of 5

Which expression represents "nine more than a number"?

Question 2 of 5

Which expression represents "four less than three times a number"?

Question 3 of 5

Which expression represents "twice the difference of a number and 7"?

Question 4 of 5

Jake earns \( d \) dollars per hour. He works 8 hours and then spends $15 on lunch. Which expression represents the amount of money Jake has left?

Question 5 of 5

A box contains \( p \) pencils. If 6 pencils are added and then the total is shared equally among 4 students, which expression gives the number of pencils each student receives?