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SSAT Middle Level: Quantitative • Algebraic Expressions and Equations

Translate a word problem into an algebraic expression.

Learn to turn stories into math using letters for unknown amounts.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 6 figures

SECTION 1

Historical Context & Motivation

Long ago, people used words to solve math puzzles. They needed a way to write unknown amounts with letters. This idea grew into algebra, which helps turn word problems into expressions.

820 AD • Birth of Algebra

Al-Khwarizmi writes the first algebra book. He uses words for unknowns.

1637 • Letters for Variables

René Descartes starts using letters like x for unknowns. This makes math clearer.

1800s • Word Problems in School

Schools teach kids to translate stories into algebra. It builds real-world math skills.

Today • SSAT Prep

You use this skill on tests like the SSAT to solve tricky problems fast.

Before algebra, math was hard for stories like "twice as many apples as friends." Now, you can write 2f. This lesson shows you how!

SECTION 2

Core Principles & Definitions

An algebraic expression is math with numbers and letters, like 3x + 2. A variable like x stands for an unknown number. To translate, match words to math symbols.

1

Addition Words

"more than", "plus", "increased by" mean +.
2

Subtraction Words

"less than", "minus", "decreased by" mean −.
3

Multiplication Words

"times", "of", "twice" mean ×.
4

Division Words

"divided by", "per", "half of" mean ÷.
5

Equals Words

"is", "equals" mean = (for equations).
✦

Key Takeaway

Think of it like a video game code. Words are the commands, and algebra is the code that makes the game work!

SECTION 3

Visual Explanation

From Words to Algebra Translating a word problem into an algebraic expression WORD PROBLEM "Twice a number minus 5" BREAK IT DOWN "Twice" means multiply by 2 "a number" unknown → use n "minus 5" subtract 5 2 × n − 5 2n − 5 ALGEBRAIC EXPRESSION 💡 Tip: Use a variable like n for any unknown number.
Arrows show how words map to symbols. The variable n holds the place of the unknown.

See how the story breaks into parts? Words like "twice" become numbers times a variable. You can do this with any problem!

SECTION 4

Mathematical Framework

Start with the unknown as a variable like n or x. Then add words step by step. Always read carefully.

COMMON TRANSLATIONS

"anumber"→n"threemorethann"→n+3"twicen"→2n"nless4"→n−4"a number" → n "three more than n" → n + 3 "twice n" → 2n "n less 4" → n − 4

n is the variable (unknown). Use × for times if needed, but often just next to number.

ORDER MATTERS

"5lessthantwiceanumber"→2n−5NOT5−2n"5 less than twice a number" → 2n − 5 NOT 5 − 2n

Read left to right: twice first, then less 5.

SECTION 5

Detailed Breakdown

Words → Symbols: Translating Phrases into Algebra Common word-problem phrases and their matching math operations ADDITION "more than" "increased by" "plus" · "sum of" + "5 more than n" n + 5 SUBTRACTION "less than" "decreased by" "minus" · "fewer" − "4 less than x" x − 4 MULTIPLICATION "times" "twice" / "triple" "product of" · "of" × "twice a number" 2n DIVISION "divided by" "split equally" "quotient" · "per" ÷ "y split into 3" y ÷ 3 PHRASES SYMBOL EXAMPLE 💡 Pro Tip: Watch for Tricky Order! "5 less than n" means n − 5 (not 5 − n). The number comes first! "3 more than twice x" becomes 2x + 3 — combine operations step by step. identify keyword write expression
Phrases point to math operations. Match them to build your expression.

This map helps spot clues in problems. Practice with friends or games to get faster. You're getting great at this!

SECTION 6

Worked Example

Translate: Three times a number plus seven

  1. 1

    Step 1: Spot the unknown

    "a number" is the variable. Let it be n.
  2. 2

    Step 2: Find multiplication

    "Three times" means 3 × n or 3n.
    3n
  3. 3

    Step 3: Add the rest

    "plus seven" means + 7. So, 3n + 7.
    3n + 7

See? Break it into small steps. You just solved it like a pro!

SECTION 7

Strengths & Common Errors

MistakeWhy WrongFix It
Switching order like 5 − 2n for "twice minus 5"Words go left to right.Read carefully: 2n − 5.
Forgetting variable like "3 + 7".No unknown!Always use n or x.
Using = in expression.Expressions have no =.Save = for equations.

Avoid these traps to ace your SSAT.

★

Key takeaway

Strength: Turns confusing stories into simple math. Like decoding a secret message in a game!

SECTION 8

Connection to Equations

ExpressionEquation (Next Level)
2n + 3 (twice plus 3)2n + 3 = 11 (equals 11)
n − 4 (less 4)n − 4 = 7
3n (three times)3n = 21

Expressions are half the story. Add "equals" to make equations and solve for n. You'll learn that next!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

What does "a number" represent? A. 5 B. n C. + D. × E. =
Show answer
B is correct. "A number" means the unknown, so use a variable like n. A is a specific number, not unknown. C, D, E are operations, not unknowns.

Question 2: BASIC CALCULATION

"Five more than a number" A. 5 − n B. n + 5 C. 5n D. n − 5 E. 5 + n
Show answer
B (or E, same) is correct: "more than" means +. A and D are subtraction (wrong direction). C is times. Order can vary if clear.

Question 3: INTERMEDIATE

"Four less than twice a number" A. 4 − 2n B. 2n − 4 C. 4n − 2 D. n − 8 E. 2(n − 4)
Show answer
B is correct. Twice first (2n), then less 4 (−4). A reverses order. C wrong ops. D not twice. E distributes wrong.

Question 4: APPLIED

A rectangle's length is 3 more than width w. A. l = w − 3 B. l = 3w C. l = w + 3 D. 3w − l E. w − 3
Show answer
C is correct. "3 more than w" is w + 3. A reverses. B multiplies. D subtracts wrong. E just subtracts.

Question 5: CRITICAL THINKING

"The sum of three times a number and 2, decreased by 7" A. 3n + 2 − 7 B. (3n + 2) − 7 C. 3(n + 2 − 7) D. 3n + 2 + 7 E. 3n − 9
Show answer
B is correct. Sum of 3n and 2 is 3n + 2, then decreased by 7: (3n + 2) − 7. A same without parens (ok often). C distributes wrong. D adds. E simplifies wrong.

SUMMARY

Lesson Summary

Summary

You now know how to translate word problems into algebraic expressions. Use variables for unknowns and match words to + − × ÷.

Practice spotting phrases like "more than" for +. Break problems into steps. You're ready for SSAT success—keep practicing!

Varsity Tutors • SSAT Middle Level • Translate a word problem into an algebraic expression.