Equation Problems
Equation problems start with words, not symbols. This NSW Year 9 Core maths subtopic, under NESA outcome MA5‑EQU‑C‑01, is about choosing a pronumeral, turning the description into an equation, solving it and answering the question that was actually asked.
You will translate phrases such as doubled and increased by, less than and times into symbols, and write consecutive numbers as \(x\) and \(x + 1\), or \(x\) and \(x + 2\) for even and odd numbers. You will write perimeters of described rectangles, a taxi fare as a fixed charge plus a rate per kilometre, shares of money and concert attendances in terms of one unknown, and use the fact that a triangle's angles add to \(180^\circ\). You will also solve age problems, such as how old Jack and his mum are. It is part of Year 9 Core Equations.
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 and costs.
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: \(9 + 14 + 9 + 14 = 46\) \(\checkmark\)
Everyone is described in terms of Leo, so let \(x\) be Leo's share in dollars.
Mia gets \(3x\). Zoe gets \(x - 10\). The three shares add to \(150\).
| \(x + 3x + (x - 10)\) | \(=\) | \(150\) |
| \(5x - 10\) | \(=\) | \(150\) |
| \(5x\) | \(=\) | \(160\) |
| \(x\) | \(=\) | \(32\) |
Leo gets $32, Mia gets \(3 \times 32 =\) $96 and Zoe gets \(32 - 10 =\) $22.
Check: \(32 + 96 + 22 = 150\) \(\checkmark\)
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.