Mathematics, three units

How do you turn a word problem into an equation?

Quick answer

To turn a word problem into an equation, write down what your unknown stands for in one line. Next, translate each sentence into an algebraic expression rather than tackling the whole text at once. Set expressions equal using total or comparison clues, solve, and check your answer against the original story to reject impossible values.

The topic explained

Define your unknown on paper first

Most points lost on high school algebra exams disappear before a student writes a single equals sign. To turn a word problem into an equation without getting lost, write one clear statement before doing any math: let x equal the quantity you need to find. If an examiner cannot see what x represents, they cannot award partial credit. More importantly, two steps later you may forget whether x was someone's age today or their age four years from now.

Choosing which quantity to label makes a difference. When comparing two quantities, assign x to the smaller one whenever possible. For example, if Danny is 6 years older than Yossi, let x be Yossi's age so Danny becomes x + 6. Working with addition avoids unnecessary minus signs, which are the most common source of calculation errors.

Translate the story one sentence at a time

Do not try to build the full equation in one leap. Treat every sentence as its own phrase. Common phrases follow predictable patterns: 'older by 5' means add 5, 'three times as many' means multiply by 3, and 'in four years' means add 4 to every person mentioned in the problem, not just one.

Percentage changes work the same way. A price that increases by 20 percent becomes 1.2x, while a 25 percent discount multiplies that result by 0.75. Once each phrase is written in symbols, look for connecting words such as 'altogether', 'is', or 'totals'. That word becomes your equals sign, joining your separate expressions into one solvable equation.

Check the story and reject impossible roots

Once you find a numerical answer, test it against the written story rather than your own equation. If you mistranslated a sentence, checking against your equation only confirms your algebraic arithmetic, not your setup.

When dealing with quadratic equations, you will often get two mathematical solutions. Physical quantities such as lengths, ages, prices, and people cannot be negative. If an equation gives widths of 5 and -12, state clearly: 'width cannot be negative, so -12 is rejected.' Writing that single sentence earns full credit and guarantees you answer what the problem actually asked.

Free printable worksheet

Translating Word Problems into Equations

For 11th-grade students. Define the unknown in writing, set up an equation, solve it, and state whether any root must be rejected.

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Translating Word Problems into EquationsName:

A notebook costs $3 more than a pen. Maya bought 4 pens and 2 notebooks for $24. Define the unknown and find the cost of one pen.
Danny is 6 years older than Yossi. In 4 years, the sum of their ages will be 40. How old is Yossi today?
A jacket was discounted by 20%. The sale price is $64. What was the original price?
A rectangle has an area of 60 square meters. Its length is 7 meters longer than its width. Find its width and explain any rejected values.
An item's price rose by 20%, then fell by 25%. The final price was 90 dollars. Find the original price.

Common questions

What is the most common mistake when writing an equation from a word problem?

The most common mistake is skipping the written definition of the variable. Without defining x on paper, students frequently assign their final numerical result to the wrong person, year, or unit.

How do I know which quantity should be x?

Pick the smaller quantity or the base quantity that other items depend on. Letting the smaller quantity equal x keeps your terms positive, which prevents sign errors during simplification.

Why do teachers require writing down why a negative root was rejected?

A quadratic formula solves pure arithmetic, but a word problem models physical reality. Demonstrating that an age or distance cannot be negative shows that you understand what the numbers mean.

What should I do if a word problem has two different unknown values?

Look for a sentence that connects them, such as one being twice the other. Express both items using a single variable x before writing your equation.

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