Algebra Word Problems Explained
Word problems ask you to translate a real-world situation into mathematical language, solve it, and then translate the answer back. Students often find them harder than pure equation-solving because the translation step is unfamiliar — but once you have a reliable method, word problems become a predictable skill rather than a source of anxiety.
A General Method for Any Word Problem
Before reaching for a pencil, read the problem twice. The first reading is for overall understanding; the second is for extracting specific quantities and relationships. A four-step approach works for almost every word problem at secondary level.
Step 1: Identify what you are asked to find. State this clearly. If the problem asks for the number of adult tickets sold, write: "Let a = number of adult tickets." Naming your unknown precisely prevents confusion later.
Step 2: Express all other unknown quantities in terms of your variable. If adult tickets cost $12 and child tickets cost $7, and you know the total number of tickets is 200, then child tickets = 200 − a. You now have one variable, not two.
Step 3: Write an equation using the constraint the problem provides. The problem says total revenue was $1,950: 12a + 7(200 − a) = 1950.
Step 4: Solve and check. Solve the equation, then substitute your answer back into the original problem context to verify it makes sense. If you get a negative number of tickets, you made an error somewhere.
Translating Language into Algebra
The biggest skill in word problems is recognising which mathematical operation a word or phrase signals. This table covers the most common translations.
- Sum, total, combined, together, more than, increased by → addition (+)
- Difference, less than, reduced by, fewer, decreased by → subtraction (−)
- Product, times, of (with fractions or percents), multiplied by → multiplication (×)
- Quotient, per, divided by, ratio of → division (÷)
- Is, equals, gives, results in, was → equals (=)
Watch for traps. "Five less than x" means x − 5, not 5 − x. "A number n is three times as large as another number m" means n = 3m, not 3n = m. Reading slowly and checking the direction of each relationship prevents most errors.
Number and Age Problems
Number problems give relationships between unknown numbers and ask you to find them.
Example: "The sum of two consecutive even integers is 86. Find both integers."
Consecutive even integers differ by 2. Let the smaller integer be n; the larger is n + 2.
2n + 2 = 86, so 2n = 84, so n = 42. The integers are 42 and 44. Check: 42 + 44 = 86. Correct.
Age problems involve relationships between people's ages at different points in time. The key is recognising that if a person is currently x years old, they were x − 5 five years ago and will be x + 10 in ten years.
Example: "Maria is three times as old as her nephew. In eight years, she will be twice as old. How old is each person now?"
Let the nephew's current age be n. Maria's current age is 3n. In eight years: nephew is n + 8, Maria is 3n + 8.
3n + 8 = 2n + 16, so n = 8. Nephew is 8; Maria is 24. Check: in 8 years they are 16 and 32; 32 = 2 × 16. Correct.
Distance, Rate, and Time Problems
These problems use the relationship:
The formula rearranges to r = d/t and t = d/r. The most important thing is keeping units consistent: if speed is in km/h, time must be in hours and distance will be in kilometres.
Example (opposite directions): "Two cyclists start from the same point and ride in opposite directions. One rides at 18 km/h and the other at 24 km/h. After how many hours are they 126 km apart?"
Let time = t hours. Combined distance = 18t + 24t = 42t.
Example (catching up): "A car leaves at 9 a.m. travelling at 60 km/h. A second car leaves at 10 a.m. on the same road at 80 km/h. At what time does the second car catch the first?"
Let t = hours of travel for the second car. The first car has travelled for t + 1 hours.
When they meet, their distances are equal:
The second car travels for 3 hours, leaving at 10 a.m., so they meet at 1 p.m.
For distance problems — especially those involving two travellers — sketching a simple diagram saves many errors. Draw a number line or a road, mark starting points and directions with arrows, and label distances in terms of your variable. The visual makes it clear which quantities are equal and which are being added or subtracted.
Mixture and Concentration Problems
Mixture problems involve combining two substances of different concentrations (or prices, or strengths) to produce a mixture with a known concentration.
The governing equation is: (amount of substance in solution A) + (amount of substance in solution B) = (amount of substance in mixture).
For concentration, this translates to: (volumeA × concentrationA) + (volumeB × concentrationB) = (volumetotal × concentrationtarget).
Example: "A chemist needs 200 mL of a 35% acid solution. She has a 20% solution and a 50% solution. How many mL of each should she mix?"
Let x = mL of the 20% solution. Then 200 − x = mL of the 50% solution.
0.20x + 100 − 0.50x = 70, so −0.30x = −30, so x = 100. She needs 100 mL of each solution. Check: 0.20(100) + 0.50(100) = 20 + 50 = 70 = 35% of 200. Correct.
Work-Rate Problems
Work problems describe how fast different agents (people, machines, pipes) complete a job. The key idea: if a worker completes a job in n hours, they complete 1/n of the job per hour. Rates add when workers work together.
Example: "Pipe A fills a tank in 4 hours. Pipe B fills the same tank in 6 hours. How long does it take to fill the tank with both pipes open?"
Pipe A's rate: 1/4 tank per hour. Pipe B's rate: 1/6 tank per hour. Combined rate: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank per hour.
The tank fills in 2 hours and 24 minutes with both pipes open.
Checking Your Answer
Always substitute your answer back into the original problem, not just the equation. Substituting into the equation only checks your algebra; substituting into the problem also checks your translation. Ask: does this answer make physical sense? A negative number of people, a speed greater than light, or a time in the future when the problem describes the past — any of these signals an error in the setup.
Summary
Algebra word problems become manageable when you follow a consistent process: identify the unknown, express all other quantities in terms of it, write one equation using the problem's constraint, then solve and check. The translation step requires recognising key words that signal addition, subtraction, multiplication, and division, and watching for directional traps in phrasing. Distance problems use d = rt; mixture problems equate total substance in the components with total substance in the product; work problems add rates. Checking the answer in the original problem context — not just the equation — is the final, non-negotiable step.