A 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.
In Spanish
problema verbal (problema de texto)
Also called
story problem, real-world problem, application problem, verbal problem
Steps 2 and 6 are the ones students skip most: decide what the question asks before calculating, and check the answer against the story at the end.
Formal definition and notation
A word problem states quantities and the relationships between them in natural language and asks for an unknown quantity. Solving it means building a mathematical model (an expression, equation, inequality, table or diagram) that represents the situation, solving the model, and interpreting the result in context with appropriate units, including a check of reasonableness.
A mathematical model is the general idea of describing a real situation with math (a formula, graph or equation). A word problem is a school exercise that asks you to build and use a small model.
Quick check: 3 questions
Question 1 of 3
Which equation matches: 7 less than a number is 15?
Show answer and explanation
Answer: B. 7 less than a number means start with the number and take away 7: n − 7 = 15, so n = 22. Choice A reverses the order, C multiplies, and D adds.
Question 2 of 3
Sam reads 24 pages a day. How many days will it take him to read a 312-page book?
Show answer and explanation
Answer: A. Split the pages into groups of 24: 312 ÷ 24 = 13 days. Choice B subtracts, C adds, and D multiplies; none of them gives a number of days.
Question 3 of 3
Pens cost 3 dollars each. How many pens can you buy with 20 dollars?
Show answer and explanation
Answer: C.20 ÷ 3 = 6 remainder 2, so you can afford 6 pens, with 2 dollars left. Seven pens would cost 21 dollars, B multiplies, and you cannot buy part of a pen.
Word Problem: frequently asked questions
How do you solve a word problem step by step?
Read the whole problem once. Write down what it asks for, with units. Choose a letter for the unknown. Translate the sentences into an equation, using a sketch or table if it helps. Solve the equation. Then check the answer in the original story, ask whether it is reasonable, and answer in a complete sentence.
Do key words always tell you the operation?
No. Key words are clues, not rules. In Tom has 5 more than Ana, and Tom has 12, the word more leads to subtraction to find Ana’s 7. And less than flips the order: 3 less than x is x − 3. Always picture what is happening in the story before you choose an operation.
How can I check if my answer is reasonable?
Estimate first with rounded numbers, then compare. Put your answer back into the story and see if every fact still holds. Check the units and the size: a person’s age will not be negative, a number of buses must be a whole number, and a sale price should be less than the original price.
ExpressionAn 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 8Mathematical ModelA mathematical model represents a real situation using numbers, equations, diagrams, or graphs, well enough to make predictions. The equation C = 15 + 9m can model the cost of a subscription after m months. Every model simplifies, and knowing what a model leaves out is as important as knowing what it includes.Problem Solving · 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 2ReasonablenessReasonableness is checking whether an answer makes sense for the problem before you accept it. Estimate first to see about how big the answer should be, then check the units and the context: a negative length, 3.5 people or 800 dollars for one pencil is not reasonable. A correctly calculated number can still fail this check.Problem Solving · Grade 3, Grade 4, Grade 5EstimateAn estimate is an answer that is close to the exact one, and to estimate is to find an answer that is close enough, usually by rounding or using friendly numbers. For example, 49 × 21 is about 50 × 20 = 1000. Estimation helps you check whether an exact answer is reasonable.Problem Solving · Grade 3, Grade 4, Grade 5RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6SubtractionSubtraction 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 3Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5VariableA 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 8Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1System of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7In 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 1