A Four-Step Method That Works on Almost Every Word Problem
- Read it twice. The first time for the story, the second time underlining numbers and the question.
- Name the unknown. Write "Let x = …" with units. This one line prevents most wrong answers.
- Translate each sentence into maths and join them with an equals sign.
- Solve, then check against the story, not only the equation. A negative age or 2.5 people is a sign something went wrong.
The templates in this calculator follow exactly this order, so each solution reads like a model answer.
Translating Words into Symbols
| Words | Maths | Example |
|---|---|---|
| is, was, gives, equals | = | "The total is 72" → T = 72 |
| of (with a fraction or percent) | × | "30% of 50" → 0.30 × 50 |
| more than, increased by | + | "5 more than x" → x + 5 |
| less than | subtract, reversed | "5 less than x" → x − 5 |
| per, each, for every | ÷ or × | "60 miles per hour" |
| twice, double | 2 × | "twice the width" → 2w |
| consecutive integers | n, n+1, n+2 | "three in a row" |
The one that causes most trouble is less than: "5 less than x" is x − 5, but "5 is less than x" is the inequality 5 < x.
Speed, Distance and Time Problems
Everything comes from d = r × t. When two things move toward each other, their speeds add. When one chases the other in the same direction, they subtract.
Two trains are 300 miles apart and travel toward each other at 55 mph and 45 mph. When do they meet?
- Closing speed = 55 + 45 = 100 mph.
- t = 300 ÷ 100 = 3 hours.
Answer: 3 hours
A boat that goes 18 mph in still water travels 60 miles each way on a river with a 3 mph current.
- Downstream speed 18 + 3 = 21 mph, time 60 ÷ 21 = 20/7 ≈ 2.86 h.
- Upstream speed 18 − 3 = 15 mph, time 60 ÷ 15 = 4 h.
Answer: about 2.86 h down, 4 h up
60 miles at 30 mph, then 60 miles at 60 mph. Average speed?
- Times: 2 h and 1 h, so 3 h in total.
- Average = 120 ÷ 3 = 40 mph, not (30 + 60) ÷ 2 = 45.
Answer: 40 mph
Work-Rate and Mixture Problems
For work problems, think in jobs per hour. If A paints a room in 6 hours and B in 12, A does 1/6 per hour and B does 1/12. Together 1/6 + 1/12 = 1/4 of the room per hour, so they finish in 4 hours. A drain works against you: a pool that fills in 8 hours and drains in 20 hours fills in 1 ÷ (1/8 − 1/20) = 13⅓ hours.
How many litres of 20% and 60% solution make 50 L of 30% solution?
- Let x = litres of 20%: 0.20x + 0.60(50 − x) = 0.30 × 50.
- 0.20x + 30 − 0.60x = 15, so −0.40x = −15 and x = 37.5.
Answer: 37.5 L of 20% and 12.5 L of 60%
Age, Number and Coin Problems
- Consecutive integers: three consecutive integers with sum 72 are n, n+1, n+2, so 3n + 3 = 72 and n = 23. They are 23, 24, 25.
- Coins: 30 dimes and nickels worth $2.70. With d dimes, 0.10d + 0.05(30 − d) = 2.70, so d = 24 and there are 6 nickels.
- Ages: a mother is 3 times as old as her daughter, and in 5 years their ages total 66. Then 3x + 5 + x + 5 = 66, x = 14, so the mother is 42.
Money, Interest, Geometry and Ratios
The money tab covers simple and compound interest, discount with tax, markup and break-even. The geometry tab turns "the length is 3 times the width plus 2 and the perimeter is 56" into W = 6.5 and L = 21.5, and splits triangle angles in a ratio such as 2 : 3 : 4 into 40°, 60° and 80°.
Common Mistakes in Word Problems
- Averaging two speeds instead of dividing total distance by total time.
- Treating "less than" like a subtraction in the order it is written.
- Mixing units, such as minutes with hours or cm with m, in one equation.
- Solving the equation but never answering the question that was asked. If it asks for the smaller number, give the smaller one.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: keyword lists such as “of means multiply” and “total means add”. They work on easy problems and fail on the ones that are actually graded.
Underrated: writing the unknown with its unit first (“x = hours Ana drove”). Most wrong setups come from not knowing what x counts.
Plain wrong: translating left to right. “5 less than x” is x − 5, not 5 − x; “x less than 5” is 5 − x. Word order and maths order differ.
A Worked Example With Real Numbers
Two trains leave towns 450 km apart and travel towards each other at 70 km/h and 80 km/h. Let t = hours until they meet. Together they close 70 + 80 = 150 km every hour, so 150t = 450 and t = 3 hours. Check: 70 × 3 = 210 km and 80 × 3 = 240 km, and 210 + 240 = 450 km. The check step is what separates a guess from a solved problem.
Practice Word Problems with Answers
- A cyclist rides 45 km in 2.5 hours. Speed?
Answer
18 km/h - Two numbers have sum 42 and one is 6 times the other.
Answer
36 and 6 - Pens cost $3 and notebooks $2. 25 items cost $62.
Answer
12 pens and 13 notebooks - Painters A (10 h) and B (15 h) together?
Answer
6 hours
Why Students Get Stuck on Word Problems, and What a Good Calculator Should Do
Searching for help with word problems usually starts late at night with one problem that will not work out. The maths is easy once the equation exists, and building it is the hard part. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to turn the sentence into an equation and then solve it, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Not knowing which number is the unknownA rectangle has a perimeter of 30 cm and the length is twice the width, find the width↗
- Translating "less than" in the wrong orderTwo numbers add to 25 and differ by 7, find them↗
- Missing a unit or a hidden conditionA train travels 180 km in 3 hours, find its speed↗
- Solving correctly but not answering the question askedA shirt costs 40 dollars after a 20 percent discount, find the original price↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the unknown defined first
- Wants the equation set up in plain words
- Wants the answer checked against the story
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.A rectangle has a perimeter of 30 cm and the length is twice the width, find the width↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.