Problem Solving Terms

Word-problem language 8

• At 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.Grade 6, Grade 7, Algebra 1
• In 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.Grade 7, Grade 8, Algebra 1
• No 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, ≥.Grade 6, Grade 7, Algebra 1
• NonnegativeNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.Grade 6, Grade 7, Algebra 1
• Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ^+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Grade 6, Grade 7, Algebra 1
• Rounded to the NearestRounded to the nearest tells you which place value to keep when you round an answer. Nearest tenth keeps one decimal place; nearest whole number keeps none. Look at the next digit to the right: 5 or more rounds up, 4 or less rounds down. So 3.14159 becomes 3.14 to the nearest hundredth.Grade 3, Grade 4, Grade 5
• Translating Words into MathTranslating 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.Grade 6, Grade 7, Algebra 1
• UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 7/0 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3/(x − 2) is undefined at x = 2. Undefined does not mean zero: 0/7 = 0, but 7/0 has no value.Grade 6, Grade 8, Algebra 1

Strategies and checking 3

• EstimateAn 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.Grade 3, Grade 4, Grade 5
• ReasonablenessReasonableness 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.Grade 3, Grade 4, Grade 5
• Word 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.Grade 3, Grade 4, Grade 5

Formulas and models 9

• Average SpeedAverage speed is the total distance traveled divided by the total time taken. If a car goes 150 miles in 3 hours, its average speed is 150 ÷ 3 = 50 miles per hour, even if it sped up and slowed down along the way. It is not the average of the separate speeds unless each speed lasted equally long.Grade 6, Grade 7, Algebra 1
• DensityDensity is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = m/V. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.Grade 8, Algebra 1, Geometry
• Distance-Rate-Time FormulaThe distance-rate-time formula is d = rt: distance equals rate (speed) times time, for anything moving at a constant rate. Solve it for the other two letters to get r = d/t and t = d/r. Driving at 45 miles per hour for 3 hours covers 45 × 3 = 135 miles.Grade 6, Grade 7, Algebra 1
• FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Grade 5, Grade 6, Grade 7
• Formula SheetA formula sheet is a page of math formulas that a test or class gives you, so you do not have to memorize them: area and volume formulas, slope and the quadratic formula. The GED math test provides one with lines such as A = π r^2 and d = rt. You still have to choose the right formula and use it correctly.Grade 7, Grade 8, Algebra 1
• Mathematical 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.Grade 6, Grade 7, Grade 8
• Mixture ProblemsMixture problems are word problems in which two or more ingredients with different values, prices or strengths are combined. Each ingredient adds amount × value to the total, and the parts add up: 3 lb of 4-dollar nuts plus 2 lb of 9-dollar nuts make 5 lb worth 12 + 18 = 30 dollars, which is 6 dollars per pound.Algebra 1
• PrincipalThe principal is the original amount of money that is borrowed, deposited or invested, before any interest is added. In the simple interest formula I = Prt, the letter P stands for the principal. Deposit 2,000 dollars at 3% simple interest for 4 years: the principal is 2,000 dollars, the interest is 240 dollars, and the total is 2,240 dollars.Grade 7, Algebra 1
• Work Rate ProblemsWork rate problems ask how long a job takes when people or machines work together. Each worker's rate is the part of the job done per hour: 1 divided by the time alone. Rates add: 1/a + 1/b = 1/t. If one pump fills a tank in 6 hours and another in 3 hours, together they need 2 hours.Algebra 1, Algebra 2

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Updated November 2026·Reviewed by the Effortless Math team