Translating words into math means turning a sentence into an expression, equation or inequality by matching words to symbols: sum means +, difference −, product ×, quotient ÷, and is means =. For example, 5 less than twice a number becomes 2n − 5. Watch the order, because less than and subtracted from reverse it.
In Spanish
traducir palabras a expresiones matemáticas
Also called
writing expressions from words, translating word problems, math key words, verbal expressions
Translate in chunks, left to right, and give each chunk its symbol. The one place to slow down is less than, which reverses the order.
Formal definition and notation
Translation is the modeling step of a word problem: choosing a variable for each unknown quantity and writing the stated relationships as an expression, an equation or an inequality. Each operation word maps to an operation, grouping words (the sum of, the quantity) map to parentheses, and comparison words map to =, <, >, ≤ or ≥. A correct translation gives true statements when the real values are substituted.
Define the variable first, in words: let n = the number of tickets.
Order matters for subtraction and division: 5 less than n is n − 5; 5 subtracted from n is n − 5; n less than 5 is 5 − n. The quotient of a and b is ab.
The sum of, the difference of, the quantity group terms: twice the sum of n and 3 is 2(n + 3), not 2n + 3.
Is, was, will be, equals and results in become =. Phrases such as at least and no more than become inequalities.
Check by plugging in a number: if n = 10, does 5 less than twice n give 15? 2(10) − 5 = 15. Yes.
Try it: solve the translation
Interactive
This balance shows 2x + 3 = 17, the translation of three more than twice a number is 17. Undo each step on both sides to find the number.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Translating Words into Math examples, step by step
Example 1: Write an expression
Warm-up
Translate: the product of 6 and a number, decreased by 4.
Let n be the number. The product of 6 and a number is 6n.
Decreased by 4 means subtract 4 from it: 6n − 4.
Check with n = 3: the product is 18, decreased by 4 is 14, and 6(3) − 4 = 14.
Answer: The expression is 6n − 4 (for n = 3 it equals 14).
Example 2: Write and solve an equation
Core
The sum of a number and 8 is three times the number. Find the number.
Let n be the number. The sum of a number and 8: n + 8.
Is three times the number: = 3n. Equation: n + 8 = 3n.
Subtract n: 8 = 2n, so n = 4.
Check: 4 + 8 = 12 and 3 × 4 = 12.
Answer: The number is 4.
Example 3: A real-world translation
Test-style
A plumber charges a 45 dollar visit fee plus 30 dollars per hour. A bill came to 165 dollars. How many hours did the plumber work?
Let h be the hours. 30 dollars per hour for h hours is 30h.
Plus a 45 dollar fee, and the bill is 165: 45 + 30h = 165.
Subtract 45: 30h = 120. Divide by 30: h = 4.
Answer: The plumber worked 4 hours.
Common mistakes
✗ Mistake7 less than a number: 7 − n.
✓ Insteadn − 7. Start with the number, then take 7 away.
Less than means start from the second quantity and subtract the first. Test it: 7 less than 10 is 3, and 10 − 7 = 3.
✗ MistakeTwice the sum of a number and 5: 2n + 5.
✓ Instead2(n + 5).
The sum of names a group that is found first. Without parentheses, only n is doubled.
✗ MistakeThe quotient of a number and 4: 4n.
✓ Insteadn4.
In the quotient of A and B, A is divided by B, in the order they are named.
A phrase with no verb, such as 4 more than x, translates to an expression, x + 4. A sentence with is or equals translates to an equation, such as x + 4 = 9, which you can solve.
A word problem is the whole task. Translating is the first step of it; solving and checking the answer come after.
Quick check: 3 questions
Question 1 of 3
Which expression means 9 less than a number x?
Show answer and explanation
Answer: B. Start with x and take away 9: x − 9. 9 − x reverses the order, 9x multiplies, and x < 9 translates is less than, not less than as a subtraction.
Question 2 of 3
Four times the difference of a number and 2 is 28. What is the number?
Show answer and explanation
Answer: C.4(n − 2) = 28, so n − 2 = 7 and n = 9. 7.5 comes from 4n − 2 = 28, and the others drop the parentheses or the subtraction.
Question 3 of 3
A gym charges 20 dollars to join plus 15 dollars per month. Which equation finds the months m for a total of 125 dollars?
Show answer and explanation
Answer: D. The one-time 20 dollars is added once and 15 dollars is multiplied by the months: 20 + 15m = 125, giving m = 7. A swaps the fee and the rate, and B and C charge both amounts every month.
Translating Words into Math: frequently asked questions
What are the key words for translating words into math?
Addition: sum, plus, more than, increased by, total. Subtraction: difference, minus, less than, decreased by, fewer. Multiplication: product, times, twice, double, of. Division: quotient, divided by, per, ratio of. Equals: is, was, gives, results in. Inequalities: at least, no more than, greater than, fewer than.
Why does less than switch the order?
Because it describes starting from one quantity and going down. 3 less than 10 means start at 10 and go down 3: 10 − 3 = 7. So x less than y is y − x. A quick test with real numbers catches the switch. Note that is less than, with the verb is, is the inequality < instead.
How do I check that my translation is right?
Pick an easy value for the variable and read the original sentence with it. If the sentence and your expression give the same number, the translation is right. For twice the sum of a number and 5, try n = 1: the sum is 6 and twice that is 12, and 2(1 + 5) = 12.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.EE.A.2.a, CCSS 7.EE.B.4
Cite this entry
Effortless Math Team. “Translating Words into Math.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/translating-words-into-math/
Flashcard
TermTranslating Words into Math5 less than twice n
DefinitionTurn each word chunk into a symbol: sum +, less than − (order flips), product ×, is =. 5 less than twice n is 2n − 5.
TranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryVariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3No More ThanNo more than means less than or equal to, written ≤. The limit itself is allowed: you may spend no more than 20 dollars means 20 dollars or any smaller amount, so x ≤ 20. It means the same as at most. The reverse phrase, no less than, means greater than or equal to, ≥.Problem Solving · Grade 6, Grade 7, Algebra 1In Terms OfIn terms of means the answer should be written using a given variable or symbol. “Write the perimeter in terms of w” wants an expression with w, such as 4w + 10. “Solve for y in terms of x” means get y alone, with only x on the other side. “In terms of π” keeps π as a symbol: 9π, not 28.26.Problem Solving · Grade 7, Grade 8, Algebra 1Word ProblemA word problem describes a math situation in everyday language instead of only symbols. To solve one, identify what the problem is asking you to find, translate the words into an expression or equation, solve it, and check that the answer makes sense in the story. It is also called a story problem or a real-world problem.Problem Solving · Grade 3, Grade 4, Grade 5AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5At Least and At MostAt least and at most are word-problem phrases for limits that include the boundary. “At least 3” means 3 or more, written x ≥ 3. “At most 3” means 3 or less, written x ≤ 3. In both cases the number 3 itself is allowed, so it gets a filled dot on a number line.Problem Solving · Grade 6, Grade 7, Algebra 1