What is a word problem?
A word problem is a short story with numbers, and one or more numbers are hidden. Our job is to find them.
Arithmetic can find the answer by trying numbers. Algebra is faster and safer: we give the hidden number a letter (like x) and then work with it like any other number. The letter is called the unknown or variable.
Look at the 3D balance above. Mia's pencils are a blue stack of x blocks. The left pan holds all the pencils, the right pan holds 13. Equal pans mean a true equation.
The algebraic method in four steps
- Name the unknown. Write exactly what it means, for example: let x = the number of Mia's pencils. Do not forget the unit.
- Equation. Say the story in maths. Words like more than mean +, less than mean −, times or twice mean ×, is or total often means =.
- Solve. Do the same thing to both sides until x is alone.
- Check. Put the answer into the story, not only into your equation. Then write the answer in a full sentence with the unit.
Translate carefully: 5 more than x is x + 5, but 5 less than x is x − 5 and x less than 5 is 5 − x.
Common kinds of word problems
Number and age problems
Two consecutive numbers: x and x + 1. Three consecutive even numbers: x, x + 2, x + 4. For ages, remember that after n years everyone is n years older.
Motion problems
distance = speed × time. If two travellers move toward each other, their speeds add. On a river, a boat goes at (boat speed + river speed) downstream and (boat speed − river speed) upstream.
Work problems
If a tap fills a tank in 6 hours, in 1 hour it fills 1/6 of the tank. Add the parts done in one hour and find the whole time.
Percent and mixture problems
A part is a percent of the whole: part = (p/100) × whole. Add the amounts of the pure substance in each mix to build the equation.
Shape problems
Use area = length × width, perimeter = 2(length + width). The story often gives how one side is related to the other.
Two unknowns and quadratic equations
Sometimes a story has two hidden numbers, like the price of a pen and the price of a notebook. Then we use two letters and write two equations. We solve them by adding, subtracting or substituting (a system of linear equations).
Sometimes the equation has x². For example, a rectangle has length (w + 3) and area 40, so w(w + 3) = 40, which gives w² + 3w − 40 = 0. Solving gives two roots, but the story may allow only one. A length cannot be negative, so we reject that root. Always test each root in the story.
Try it: a balance at home
Take a ruler and a pencil as a see-saw, two small cups and 20 identical coins or beans. Put some beans in a cup and hide them under a cloth: that number is x. Ask a friend to put the same cup on the other side and add 3 beans on one side. Predict first: how many beans are in the cup if the see-saw stays level when the other side has 13? Then open the cloth and check. Write the equation you used.
Key formulas and definitions
- Steps: Name the unknown → Equation → Solve → Check
- Consecutive numbers: x, x + 1, x + 2
- distance = speed × time
- Meeting towards each other: (speed A + speed B) × time = distance
- River: downstream = boat + river, upstream = boat − river
- Work: parts per hour add up; time together = 1 ÷ (1/a + 1/b)
- Rectangle: area = length × width, perimeter = 2(length + width)
Worked examples
1. The sum of two consecutive numbers is 41. Find them.
Let the smaller number be x. The next is x + 1. Equation: x + (x + 1) = 41, so 2x + 1 = 41, 2x = 40, x = 20. The numbers are 20 and 21. Check: 20 + 21 = 41.
2. A father is 4 times as old as his son. In 5 years the sum of their ages will be 55. Find their ages now.
Let the son be x years, the father 4x. In 5 years: (x + 5) + (4x + 5) = 55, so 5x + 10 = 55, 5x = 45, x = 9. Son 9, father 36. Check: in 5 years 14 + 41 = 55.
3. Two towns are 180 km apart. A car leaves each town at the same time, driving toward each other at 50 km/h and 40 km/h. When do they meet?
Let the time be t hours. Distances: 50t + 40t = 180, so 90t = 180, t = 2. They meet after 2 hours. Check: 100 km + 80 km = 180 km.
4. A tap fills a tank in 6 hours. A second tap fills it in 3 hours. How long do they take together?
In 1 hour the taps fill 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 of the tank. So the whole tank takes 2 hours.
5. 3 pens and 2 notebooks cost Rs 70. 2 pens and 3 notebooks cost Rs 80. Find the price of a pen and a notebook.
Let a pen cost p and a notebook n. 3p + 2n = 70 and 2p + 3n = 80. Subtract the second from the first: p − n = −10. Add them: 5p + 5n = 150, so p + n = 30. Then n = p + 10, so 2p + 10 = 30, p = 10, n = 20. A pen costs Rs 10 and a notebook Rs 20. Check: 30 + 40 = 70 and 20 + 60 = 80.
6. A rectangle is 3 m longer than it is wide. Its area is 40 m². Find its sides.
Let the width be w m. Length w + 3. Area: w(w + 3) = 40, so w² + 3w − 40 = 0. Factor: (w + 8)(w − 5) = 0, so w = −8 or w = 5. A width cannot be negative, so w = 5 m and the length is 8 m. Check: 5 × 8 = 40.
7. A boat goes 36 km downstream and 36 km back upstream in 9 hours in total. The river flows at 3 km/h. Find the speed of the boat in still water.
Let the boat's speed be v km/h. Time = distance ÷ speed, so 36/(v + 3) + 36/(v − 3) = 9. Multiply by (v + 3)(v − 3): 36(v − 3) + 36(v + 3) = 9(v² − 9), so 72v = 9v² − 81, v² − 8v − 9 = 0, (v − 9)(v + 1) = 0. v = 9 (speed cannot be negative). Check: 36/12 + 36/6 = 3 + 6 = 9 hours.
Common mistakes
- Starting to write equations without saying what x means. Always write: let x = ... with the unit.
- Mixing up 'less than'. 5 less than x is x − 5, not 5 − x.
- Checking only in the equation. If you built the equation wrongly, it still checks. Check in the story.
- Keeping an answer that does not fit the story, such as a negative length or a part-person like 2.5 children.