Word Problems: Turning Words into Math

Translating Keywords into Operations and Relations

In algebra, an unknown quantity in a word problem is represented by a variable, commonly a letter such as x. Word problems describe mathematical relationships through everyday language, so the initial task is recognizing which terms signal arithmetic operations. Addition is indicated by words such as sum, total, increased by, and combined. The phrase more than also indicates addition, though the order of terms is typically reversed from the verbal order. Multiplication is signaled by words such as product, times, and multiplied by. Additionally, in word problems involving fractions or percentages, the word of indicates multiplication. Special scaling words also define specific factors; for example, twice means multiplying a quantity by 2, whereas triple means multiplying a quantity by 3.

Subtraction and division require careful attention to the order in which terms appear. Words such as difference, decreased by, and diminished by typically indicate subtraction. The word less indicates subtraction in phrases such as 5 less than x. However, the phrase less than indicates subtraction where the order of operands is reversed, meaning that a less than b translates algebraically to b - a. Similarly, the phrase subtracted from places the first mentioned term after the minus sign, meaning x subtracted from y translates directly to y - x. For division, the key indicator words are quotient, divided by, ratio, and per. The word per indicates a ratio or rate expressed using division, and it gives a unit rate when the second quantity is one unit.

Equations and inequalities establish relationships between these mathematical expressions. Words such as is, are, was, were, yields, and results in translate directly to an equal sign. When expressions are compared rather than equated, specific phrasing indicates inequalities. The phrase at least indicates a minimum value and is translated using the greater-than-or-equal-to symbol, while no less than is likewise translated mathematically using the greater-than-or-equal-to symbol. Conversely, the phrase at most indicates a maximum value and is translated using the less-than-or-equal-to symbol, and no more than is also translated mathematically using the less-than-or-equal-to symbol. Strict comparisons avoid equality altogether: the word exceeds translates to the strictly greater-than symbol, whereas fewer than translates to the strictly less-than symbol.