Solving Word Problems
Every question with a fully worked solution.
- Solving Word Problems - Video - Forming and solving equations from worded problems Watch
Theory
An equation problem starts as a sentence and ends as an equation. This Year 9 maths page shows how to choose a pronumeral, turn the words into symbols, solve the equation and then answer the question that was asked. It covers number puzzles, consecutive numbers, rectangles, sharing, costs and ages.
An equation problem (or word problem) describes a situation in words. You write an equation for it, then solve it.
Start by choosing a pronumeral for the thing you know least about. Write down what it stands for, such as "Let \(x\) be the width in centimetres."
Write every other quantity in terms of that pronumeral. If the length is \(5\) more than the width, the length is \(x + 5\).
Consecutive numbers follow each other, such as \(14\) and \(15\). Consecutive even or odd numbers go up in twos, such as \(17\) and \(19\).
Solve the equation, then read the question again. Give the answer in words, with units.
Turning words into symbols, with \(x\) as the number:
| In words | In symbols |
|---|---|
| a number, doubled, then \(5\) added | \(2x + 5\) |
| three times a number, less \(4\) | \(3x - 4\) |
| a number trebled, then decreased by \(1\) | \(3x - 1\) |
| two consecutive numbers | \(x\) and \(x + 1\) |
| three consecutive numbers | \(x\), \(x + 1\) and \(x + 2\) |
| two consecutive even (or odd) numbers | \(x\) and \(x + 2\) |
| two numbers that differ by \(6\) | \(x\) and \(x + 6\) |
| $20 plus $8 for each hour | \(20 + 8h\) |
How to solve an equation problem
- Let: choose a pronumeral and write down what it stands for.
- Build: write the other quantities using it, then write the equation.
- Solve the equation and check it in the original words.
- Answer the question that was asked, in words, with units.
Let \(x\) be the number. Doubled is \(2x\). Add \(5\) to get \(2x + 5\).
| \(2x + 5\) | \(=\) | \(23\) |
| \(2x\) | \(=\) | \(18\) |
| \(x\) | \(=\) | \(9\) |
Check: \(2 \times 9 + 5 = 23\) \(\checkmark\)
The number is 9.
Let \(x\) be the smaller number. The next number is \(x + 1\).
| \(x + (x + 1)\) | \(=\) | \(47\) |
| \(2x + 1\) | \(=\) | \(47\) |
| \(2x\) | \(=\) | \(46\) |
| \(x\) | \(=\) | \(23\) |
The question asks for both numbers, so add one more line: \(x + 1 = 24\).
The numbers are 23 and 24.
Let \(x\) be the width in cm. Then the length is \(x + 5\). Add all four sides.
| \(x + (x + 5) + x + (x + 5)\) | \(=\) | \(46\) |
| \(4x + 10\) | \(=\) | \(46\) |
| \(4x\) | \(=\) | \(36\) |
| \(x\) | \(=\) | \(9\) |
The length is \(9 + 5 = 14\).
Width 9 cm, length 14 cm. Check: \(9 + 14 + 9 + 14 = 46\) \(\checkmark\)
Let \(x\) be Kate's age now, in years. Her mother is \(3x\). In \(12\) years, add \(12\) to both ages: \(x + 12\) and \(3x + 12\).
| \(3x + 12\) | \(=\) | \(2(x + 12)\) |
| \(3x + 12\) | \(=\) | \(2x + 24\) |
| \(x + 12\) | \(=\) | \(24\) |
| \(x\) | \(=\) | \(12\) |
Kate is 12 and her mother is \(3 \times 12 =\) 36.
Check: in \(12\) years they will be \(24\) and \(48\), and \(48 = 2 \times 24\) \(\checkmark\)
Common pitfalls
Frequently asked questions
How do you turn a word problem into an equation?
Choose a pronumeral for the unknown and say what it stands for. Write each quantity in the problem using that pronumeral, then set up the equation from the sentence that gives a total or a result.
How do you write consecutive numbers in algebra?
Consecutive numbers are \(x\), \(x + 1\), \(x + 2\) and so on. Consecutive even or odd numbers go up in twos: \(x\), \(x + 2\), \(x + 4\).
Which unknown should be x?
Usually the one the others are described from. If Mia gets three times as much as Leo, let \(x\) be Leo's share, so Mia's is \(3x\).
What does trebled mean in maths?
Trebled means multiplied by \(3\). A number trebled is \(3x\).
Why check the answer in the words of the problem?
An equation can be solved correctly but built wrongly. Putting the answer back into the original sentence shows whether the equation matched the problem.