Word Problems

Every word problem in beginning algebra has the same shape: a few sentences describing how some quantities relate, and a question asking for one of them. Solving it is a translation task first and an algebra task second. Once the paragraph becomes an equation, the equation is solved with the ordinary techniques from Linear Equations — nothing new happens there.

This lesson is about the translation. It builds on the five-step process introduced in Applications of Linear Equations and focuses on the general skill: reading carefully, assigning one variable, and turning each relationship in the text into algebra. The specialized problem types — motion, work, and mixture — each get their own lesson, linked at the end.

The Word-Problem Formula

The arrows are the method. The two steps students skip are the second one — pinning down exactly what means — and the last one — turning the solved value back into an answer to the question that was actually asked.

The Five-Step Method

1. Read it twice and identify the question. Restate what the problem is asking for, in your own words, before writing anything.
2. Assign one variable. Let be the simplest unknown. Write every other unknown quantity as an expression in .
3. Translate a relationship into an equation. Find the sentence that says two things are equal (or that one total is made of parts) and write it in symbols.
4. Solve the equation with the methods from Linear Equations.
5. Answer the question and check it against the words. Attach units, write a sentence, and confirm the answer makes physical sense.

Translating Words into Symbols

Two habits matter more than memorizing the table. First, “less than” and “decreased by” reverse the order you read them in: “8 less than a number” is , not . Second, a phrase with a comma or an “and then” usually needs parentheses: “a number decreased by 5, then tripled” is , because the tripling applies to the whole decreased quantity.

Number Problems

The most direct type: one or two unknown numbers with a stated relationship and a stated total or equation.

When a problem mentions a “first number” and a “second number,” let be the first and write the second in terms of it, then use the relationship the problem gives to form one equation.

Age Problems

Age problems hinge on one fact: everyone ages at the same rate. If a person is years old now, they were years old four years ago and will be years old in ten years. Set up “now” ages first, then adjust both people by the same amount for the past or future condition.

Consecutive Integer Problems

Consecutive integers follow one after another: . Consecutive even integers and consecutive odd integers both skip by two: . One variable covers the whole run.

Part-of-a-Whole Problems

When a fixed total is split into pieces described in terms of each other, let be one piece, express the others in terms of , and set the sum equal to the total.

Worked Example A: A Number Problem

The sum of three times a number and 7 is 34. Find the number.

Step 1 — the question: find the number.

Step 2 — assign a variable. Let be the number.

Step 3 — translate:

Step 4 — solve:

Step 5 — answer and check. The number is . Check: . ✓

Worked Example B: An Age Problem

Maria is 3 times as old as her son. In 12 years, she will be twice as old as he is then. How old is each now?

Step 1 — the question: the current age of each.

Step 2 — assign a variable. Let be the son’s age now; Maria is now. In 12 years the son is and Maria is .

Step 3 — translate the future condition:

Step 4 — solve:

Step 5 — answer and check. The son is and Maria is . In 12 years they are and , and . ✓

Worked Example C: Consecutive Integers

The sum of four consecutive integers is 90. Find them.

Step 1 — the question: the four integers.

Step 2 — assign a variable. Let be the smallest; the four are .

Step 3 — translate:

Step 4 — solve:

Step 5 — answer and check. The integers are , and . ✓

Worked Example D: Part of a Whole

A collection of 48 coins is all nickels and dimes. There are 6 more dimes than nickels. How many of each are there?

Step 1 — the question: the number of nickels and the number of dimes.

Step 2 — assign a variable. Let be the number of nickels; the number of dimes is .

Step 3 — translate the total count:

Step 4 — solve:

Step 5 — answer and check. There are nickels and dimes, and . ✓

Common Mistakes to Avoid

• Using a different letter for every unknown. Two letters need two equations; one variable with expressions needs only one.
• Reversing a subtraction. “7 less than ” is . Reading it left to right as is the most common translation error there is.
• Forgetting parentheses after a comma. “The sum of a number and 4, doubled” is , not .
• Solving for and stopping. If the question asks for the largest integer or the number of dimes, itself is usually not the final answer.
• Aging only one person. In an age problem, every person’s age changes by the same amount for the same time shift.
• Accepting a nonsense answer. A negative count, a fractional person, or an impossible length means the equation is wrong — return to step 3.

Where This Shows Up Later

• Distance, Rate, and Time. The same five steps, applied with the relationship .
• Work Problems. The same five steps, applied with combined work rates that add.
• Mixture Problems. The same five steps, applied with (amount)(concentration) totals.
• Quadratic Applications. Area, projectile, and product problems use this exact routine but end in a quadratic equation.
• Systems of Equations. Problems with two genuinely independent unknowns are set up as two equations instead of forcing one variable.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Five more than twice a number is 23. Find the number.

Problem 2. Seven less than four times a number equals the number increased by 8. Find the number.

Problem 3. The sum of two consecutive integers is 71. Find them.

Problem 4. The sum of three consecutive even integers is 84. Find them.

Problem 5. A father is 4 times as old as his daughter. In 20 years he will be twice as old as she is then. Find their current ages.

Problem 6. One number is 5 more than another. Their sum is 33. Find both numbers.

Problem 7. A 72-inch ribbon is cut into two pieces so that one piece is 3 times the length of the other. Find both lengths.

Problem 8. The sum of three consecutive odd integers is 129. Find them.

Problem 9. Twice a number, decreased by 9, is the same as the number increased by 6. Find the number.

Problem 10. A jar has 40 marbles, all red or blue. There are 4 fewer blue marbles than twice the number of red marbles. How many of each are there?

Problem 11. The larger of two numbers is 1 less than 3 times the smaller. Their sum is 27. Find both numbers.

Problem 12. Sam is 6 years older than Kim. Four years ago, Sam was twice as old as Kim was then. Find their current ages.

Quick Reference

Once the equation is written, everything after that is Linear Equations. For the three big specialized types, see Distance, Rate, and Time, Work Problems, and Mixture Problems. The geometry and interest applications live in Applications of Linear Equations. More of the Algebra lessons are available too.