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\(x+5\)

Situations to
Expressions

Learn to translate real-world situations into algebraic expressions using variables, operations, and mathematical reasoning.

ISEE Middle Level

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables (letters like \( x \) or \( n \)), and operations (such as \( + \), \( - \), \( \times \), \( \div \)) that represents a quantity.

Expression
\( 3x + 7 \)
Equation
\( 3x + 7 = 22 \)

The Building Blocks

Variable
A letter that stands for an unknown number: \( x \), \( n \), \( t \)
Constant
A fixed number that doesn't change: \( 5 \), \( 12 \), \( -3 \)
Operation
What you do to the numbers: add, subtract, multiply, divide

Why Translate Words into Math?

The ISEE tests whether you can take a word problem and convert it into a math expression. This skill is the bridge between reading a problem and solving it.

Situation (words) Translate Expression (algebra)

The Translation Dictionary

Certain English words map to specific math operations. Memorize these keyword families:

Operation Keywords Symbol
Addition sum, more than, increased by, plus, total \( + \)
Subtraction difference, less than, decreased by, fewer \( - \)
Multiplication product, times, of, each, twice, triple \( \times \)
Division quotient, per, divided by, ratio, split \( \div \)

Order Matters: Subtraction and Division

Addition and multiplication are commutative: you can swap the order. But subtraction and division are not. The phrase structure tells you the order.

✓ Correct
"5 less than \( x \)" means \( x - 5 \). The amount after "less than" comes first.
✕ Incorrect
"5 less than \( x \)" does not mean \( 5 - x \). That reverses the order.

How to Translate Step-by-Step

1
Identify the unknown. What quantity is changing or unspecified? Assign it a variable.
2
Spot the keywords. Circle or underline words that signal an operation (sum, product, less than, etc.).
3
Determine the order. For subtraction and division, identify which quantity comes first.
4
Build the expression. Combine the variable, constants, and operations into one expression.
5
Check with a number. Plug in a simple value (like 10) and see if the expression matches the original situation.

Example 1: Simple Addition

\(s\)
 stickers she has
\(+\)
\(12\)
 more stickers
\(s + 12\)

Example 2: Multiply then Subtract

\(n\)
 a number tripled
\(3n\)
 minus 8
\(3n - 8\)

Example 3: Division with Context

1
Identify the unknown: total dollars → \( d \)
2
Split equally: Jake + 3 friends = 4 people, so each gets \( \frac{d}{4} \)
3
Donates \$2: subtract 2 from each share → \( \frac{d}{4} - 2 \)

Special Words: Twice and Half

Some keywords represent specific multipliers. Recognizing them immediately saves time on the ISEE.

Phrase Means Expression
Twice a number \( 2 \times n \) \( 2n \)
Half of a number \( n \div 2 \) \( \frac{n}{2} \)
Triple, then add 4 \( 3 \times n + 4 \) \( 3n + 4 \)
A third of a number, minus 1 \( n \div 3 - 1 \) \( \frac{n}{3} - 1 \)

When to Use Parentheses

Some situations require grouping with parentheses because the order of operations changes the result.

Without Parentheses
"Add 5 to a number, then multiply by 3" could be misread as \( n + 5 \times 3 = n + 15 \).
With Parentheses
The correct expression is \( 3(n + 5) \), because you add first, then multiply.

Example 4: Grouping in Action

sum (do first)
\(n\)
\(+\)
\(7\)
\(2\)
\(\times\)
\(2(n+7)\)

Common Mistakes to Avoid

✕ Reversing Subtraction
Writing \( 5 - x \) for "5 less than \( x \)." Correct: \( x - 5 \). The number after "less than" goes first.
✕ Forgetting Parentheses
Writing \( 2n + 7 \) for "twice the sum of \( n \) and 7." Correct: \( 2(n + 7) \). Group first, then multiply.
✕ Miscounting People
Dividing by 3 when "you and 3 friends" means 4 people total. Always count the subject of the sentence too.

Quick Reference Summary

English Phrase Expression
A number increased by 9 \( n + 9 \)
7 less than a number \( n - 7 \)
The product of 4 and a number \( 4n \)
A number divided by 6 \( \frac{n}{6} \)
3 times the sum of a number and 2 \( 3(n + 2) \)
5 more than twice a number \( 2n + 5 \)
Half a number, decreased by 4 \( \frac{n}{2} - 4 \)

Question 1 of 5

Which expression represents "8 less than a number \( n \)"?

Question 2 of 5

Lena earns \( d \) dollars per hour. She works for 5 hours and then spends \$12. Which expression represents the money she has left?

Question 3 of 5

Which expression represents "four times the sum of \( x \) and 3"?

Question 4 of 5

A pizza with \( s \) slices is shared equally among you and 4 friends. Which expression shows how many slices each person gets?

Question 5 of 5

Kai has twice as many cards as Mia. If Mia has \( m \) cards, and Kai gives away 6 cards, which expression shows how many cards Kai has now?