Word Problems

A full word problem often bundles several conditions and several possible unknowns into a single paragraph. Solving it well means breaking the paragraph into separate conditions, choosing a variable that keeps the resulting equations simple, and — at the very end — making sure the number reported actually answers the question that was asked, not just the variable that was solved for.

What you will learn

• Translate a multi-condition word problem into equations
• Choose variables that keep the equations simple
• Solve and interpret the result in the problem's original terms

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Breaking a paragraph into conditions

A multi-sentence word problem is really a list of separate conditions joined together in prose. Reading it one sentence at a time, and turning each sentence into its own short equation, keeps the translation manageable even when the whole paragraph looks complicated. Phrases like "more than," "less than," "twice as many as," and "the sum of" each have a direct algebraic translation, and mixing up their order is the most common source of an incorrect equation.

"Sam has 20 dollars more than Priya" becomes $\text{Sam}=\text{Priya}+20$ — the quantity that comes right after "more than" is the one being added to, not the one doing the adding. Reading the sentence slowly, phrase by phrase, in the order it is written prevents this equation from accidentally being flipped.

Choosing a variable that keeps equations simple

A word problem often allows more than one reasonable choice of variable, and some choices lead to much simpler equations than others. If a condition is phrased in terms of a total — "a friend received 3 more than half of the total" — letting the variable stand for the total itself avoids introducing a fraction that would otherwise appear when solving backward from a part to the whole.

As a rule, look at which quantity the problem's conditions are phrased directly in terms of, and let that quantity be the variable; the other unknowns can then usually be written as simple expressions in that variable, without any division. When a condition is a comparison such as "7 fewer than twice the number of child tickets," letting the variable be the child ticket count lets the adult count be written directly as $2c-7$, with no fraction required.

Interpreting the solution in the problem's own terms

Solving the equation produces a value for whichever variable was chosen, but that variable is not always the quantity the question actually asked for. If the variable represents Priya's money but the question asks how much Sam has, the final answer requires one more step: substituting the solved value back into the expression for Sam's money, not simply reporting the variable's value.

A finished answer should also be sanity-checked against the real-world meaning of the problem: counts of people, tickets, or packages should be non-negative whole numbers, and a reported answer that is negative or fractional when the context requires a whole count is a signal that a sentence was mistranslated somewhere earlier.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Flipping a comparison phrase, such as writing $\text{Priya}=\text{Sam}+20$ instead of $\text{Sam}=\text{Priya}+20$.
• Reporting the value of the variable that was solved for, even when the question asked about a different quantity built from it.
• Choosing a variable for a quantity that is only ever described indirectly, introducing avoidable fractions into every equation.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Translate a word problem sentence by sentence, reading comparison phrases in the exact order they are written.
• Choose a variable for whichever quantity the problem's conditions are phrased directly in terms of, so the other unknowns avoid fractions.
• Substitute the solved variable back into the expression the question actually asks about, and check that the final answer is sensible in context.