Introduction

Translating everyday language into algebra is one of the most important early skills in mathematics.
In this article, you’ll learn how to turn short word problems into clear algebraic expressions using the ideas you already know about variables and constants.

Why We Translate Word Problems

• Word problems describe relationships using ordinary language.
• Algebraic expressions describe the same relationships using symbols.
• Translating between the two helps you:
  • Understand problems more clearly
  • Prepare for solving equations later
  • Communicate mathematical ideas efficiently

Common Words and Their Algebraic Meanings

Here are some everyday phrases and the algebra they usually correspond to:

Addition

• “more than” → $x + 5$
• “increased by” → $a + b$
• “the total of” → $m + n$

Subtraction

• “less than” → $x - 3$
• “decreased by” → $p - q$
• “the difference between” → $a - b$

Multiplication

• “twice”, “double” → $2x$
• “three times” → $3k$
• “the product of” → $ab$

Division

• “split equally among” → $x/4$
• “the ratio of” → $a/b$

Other Useful Phrases

• “a number” → use a variable like $x$
• “the same number” → use the same variable
• “consecutive numbers” → $n$, $n+1$, $n+2$

Step-by-Step Translation Strategy

To translate a word problem into an expression:

1. Identify the unknown quantity
   • Choose a variable like $x$ or $n$.
2. Find the operations described
   • Look for keywords such as “more than”, “twice”, “difference”.
3. Build the expression piece by piece
   • Keep the structure simple.
   • Use parentheses when needed.
4. Check that the expression matches the story
   • Ask: “Does this expression increase/decrease the way the sentence describes?”

Examples

Example 1: Saving Money

Word problem:
“Sarah has \$15. She saves \$3 each week. Write an expression for the total amount of money she has after $w$ weeks.”

How to translate it:

• Fixed amount she already has → $15$
• Amount added each week → $3$
• Number of weeks → $w$
• Total amount → starting amount + weekly savings

Expression: $$15 + 3w$$

Example 2: Buying Snacks

Word problem:
“A granola bar costs \$2. If you buy $n$ granola bars, how much do you spend?”

How to translate it:

• Cost per bar → $2$
• Number of bars → $n$
• Total cost → cost per bar × number of bars

Expression: $$2n$$

Example 3: Temperature Change

Word problem:
“The temperature is $t$ degrees. It drops by 7 degrees overnight. Write an expression for the new temperature.”

How to translate it:

• Starting temperature → $t$
• Drop of 7 degrees → subtract 7

Expression: $$t - 7$$

Example 4: Sharing Equally

Word problem:
“A teacher has $x$ stickers and wants to give them equally to 4 students. Write an expression for how many stickers each student gets.”

How to translate it:

• Total stickers → $x$
• Shared equally among 4 → divide by 4

Expression: $$x/4$$

Example 5: Combining Several Steps

Word problem:
“A number is doubled, then 5 is added. Write an expression for the result.”

How to translate it:

• Unknown number → $n$
• Doubled → $2n$
• Then add 5 → $2n + 5$

Expression: $$2n + 5$$

Exercises

1. Translate: “A number increased by 9.”

   Solution

   $x + 9$
2. Translate: “Twice a number, then add 3.”

   Solution

   $2x + 3$
3. Translate: “The difference between a number and 12.”

   Solution

   $x - 12$
4. Translate: “Half of a number.”

   Solution

   $x/2$
5. Translate: “The total of a number and three consecutive numbers.”

   Solution

   $n + (n+1) + (n+2) + (n+3)$
6. Translate: “Five dollars less than the cost $c$.”

   Solution

   $c - 5$
7. Translate: “The product of 7 and a number.”

   Solution

   $7x$
8. Translate: “A number $k$ divided by the sum of 2 and 3.”

   Solution

   $k/5$