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\( x \)

Variables in
Context

Learn to interpret what variables represent in real-world problems and translate between words and algebraic expressions.

ISEE Middle Level

What Is a Variable?

A variable is a letter or symbol that stands for a number we don't know yet. It acts as a placeholder in an expression or equation.

\( n \)
Variable
 "the number of students"
\( n = 25 \)
\( n = 30 \)
Possible values

Variables Tell a Story

On the ISEE, variables almost always mean something in the real world. The key skill is reading the problem and figuring out what each letter represents.

\( t \)
"the number of hours Tara studied"
\( p \)
"the price, in dollars, of one book"
\( d \)
"the total distance, in miles, he biked"

From Words to Expressions

Every word problem hides a mathematical relationship. Translating English into algebra is like learning a second language: certain words always map to certain operations.

"5 more than a number"
\( n + 5 \)
 "twice the price"
\( 2p \)

The Translation Dictionary

English Phrase Operation Algebra
"sum of," "more than," "increased by" Addition \( x + k \)
"difference," "less than," "decreased by" Subtraction \( x - k \)
"product of," "times," "twice," "triple" Multiplication \( k \cdot x \)
"quotient of," "divided by," "per" Division \( \frac{x}{k} \)
"is," "equals," "the result is" Equals sign \( = \)

Variable vs. Constant

In the expression \( 3n + 7 \), there are two kinds of pieces. Understanding the difference is critical for interpreting what a problem is really asking.

Variable: \( n \)
Changes depending on the situation. Could be 1, 10, or 100. It represents an unknown quantity.
Constants: 3 and 7
Fixed numbers that never change. The 3 is a coefficient (multiplier of \( n \)) and the 7 is a standalone constant.

Reading Expressions in Context

When a problem gives you an expression like \( 4s + 12 \), each piece tells part of the story. Here is a method to decode any expression.

1
Identify the variable. What real-world quantity does the letter represent?
2
Read the coefficient. What is being multiplied by the variable? This is the rate or per-unit amount.
3
Read the constant. What number is added or subtracted? This is the starting value or fixed amount.
4
State the meaning. Put it all together in a complete sentence.

Step-by-Step Strategy

1
Read the whole problem. Underline what each variable stands for.
2
Highlight signal words: "total," "each," "per," "more than," "less than."
3
Match each part of the expression to a piece of the story.
4
Plug in a test number. If the problem says \( n = 3 \) books, does \( 8n \) give $24? Does that make sense for "8 dollars per book"?
5
Check the answer choices. Eliminate any that contradict the context.

Example 1: Movie Tickets

$9 $9 $9 ...
\( t \) tickets
\( 9t \)
total dollars

Example 2: Savings Account

Weeks (\( w \))
 Dollars ($) 0 1 2 3 4 50 65 80 95 110 $50 start +1 wk +$15

Example 3: Pizza Party

\( p \)
Number of pizzas ordered
\( 12p \)
Cost of the pizzas alone (before delivery)
5
Fixed delivery fee, same no matter how many pizzas

Expressions with Multiple Variables

Some problems use more than one variable. Each letter represents a different unknown quantity.

\( 3s \)
Cost of 3 shirts (rate \( s \) per shirt)
\( 2p \)
Cost of 2 pants (rate \( p \) per pair)

What Changes and What Stays Fixed?

A powerful way to interpret expressions is to ask: which part grows when the variable increases, and which part stays the same?

2.50
\( m \)
 + 3 Changes with each mile driven Fixed base fare ($3) $2.50 per mile

Common Mistakes

✕ Reversing Subtraction
Writing \( 5 - n \) instead of \( n - 5 \) for "5 less than \( n \)." The number being subtracted from is the variable.
✕ Ignoring Context
Solving for \( x \) without checking what \( x \) actually represents. Your answer might be correct numerically but wrong in meaning.
✕ Confusing Coefficient and Constant
In \( 7n + 4 \), the 7 is the rate (per-unit), not a starting amount. The 4 is the fixed starting amount.

Summary Reference

Part of Expression What It Means Example in \( 6h + 20 \)
Variable (\( h \)) Unknown quantity that changes Hours worked
Coefficient (6) Rate per unit of the variable $6 per hour
Constant (20) Fixed amount, does not change $20 signing bonus
Entire expression Total outcome combining all parts Total earnings ($)

Question 1 of 5

A gym charges $25 per month plus a $40 registration fee. If \( m \) represents the number of months, what does the expression \( 25m + 40 \) represent?

Question 2 of 5

In the expression \( 8n \), if \( n \) represents the number of books a student buys, what does 8 most likely represent?

Question 3 of 5

Which expression represents "7 less than twice a number \( x \)"?

Question 4 of 5

A car rental costs $35 per day plus $0.20 per mile. If \( d \) is the number of days and \( m \) is the number of miles, what does \( 0.20m \) represent?

Question 5 of 5

Jenna earns \( 10h + 15 \) dollars, where \( h \) is the number of hours she works. If she works 0 hours, she still earns $15. What does the 15 most likely represent?