Equation Problems
Equation problems turn words into algebra. In NSW Year 8 maths (NESA outcome MA4‑EQU‑C‑01) you name the unknown with a pronumeral, write every other amount in terms of it, form an equation, solve it, and then answer the question in a sentence with units.
You will solve number puzzles, problems about consecutive and consecutive even numbers, ages, perimeters and sharing the money raised at a car wash, and substitute into formulas such as a phone battery's charge after \(h\) hours. When two amounts are compared, such as two gym memberships, the pronumeral ends up on both sides. Each answer is checked against the words of the problem. It is part of Year 8 Equations.
Theory
Name the unknown, write every other quantity with that same pronumeral, form an equation, solve it, then answer the question in words. For two consecutive odd numbers adding to \(136\): \(x + (x + 2) = 136\), so the numbers are \(67\) and \(69\). This Year 8 maths page shows how.
1. Define it in a sentence, with units. “Let \(n\) be the number of tickets.” “Let \(d\) be the distance in kilometres.”
2. Name the other quantities with the same pronumeral. If the long piece is \(4\) times the short piece \(x\), it is \(4x\). If a price is \(\$7\) dearer, it is \(x + 7\). Consecutive numbers are \(x\) and \(x + 1\); consecutive even (or odd) numbers are \(x\) and \(x + 2\).
3. Write the equation. Words such as is, gives, leaves, costs and altogether become \(=\). If the problem compares two amounts, the pronumeral lands on both sides.
4. Solve it, collecting like terms first, then one inverse operation per line.
5. Answer in words, with units, and answer the question that was asked. 6. Check in the words of the problem, not just in your own equation.
Ages. In \(n\) years, every age goes up by \(n\): if Tom is \(x\) now, he will be \(x + 15\) in \(15\) years.
Formulas. Write the formula, substitute every value you know, then solve for the pronumeral that is left.
Words into equations:
| Words | Equation |
|---|---|
| \(8\) is taken from \(5\) times a number, leaving \(32\). | \(5n - 8 = 32\) |
| A number is doubled and \(7\) added, giving \(19\) more than the number. | \(2n + 7 = n + 19\) |
| Tickets cost \(\$18\) each, and a \(\$7.50\) fee makes the total \(\$97.50\). | \(18t + 7.50 = 97.50\) |
| A taxi charges \(\$4\) plus \(\$2.20\) a kilometre, and the fare is \(\$37\). | \(4 + 2.20k = 37\) |
How to solve a worded problem
- Let a pronumeral stand for the unknown, with units.
- Write the other quantities with the same pronumeral.
- Form the equation and solve it.
- Answer the question in words, with units, and check in the story.
Let \(n\) be the number. Trebled means \(\times 3\).
| \(3n + 6.5\) | \(=\) | \(27.5\) |
| \(3n\) | \(=\) | \(21\) |
| \(n\) | \(=\) | \(7\) |
The number is \(7\). Check: \(3 \times 7 + 6.5 = 27.5\).
Let the smaller number be \(x\). The next odd number is \(x + 2\).
| \(x + x + 2\) | \(=\) | \(136\) |
| \(2x + 2\) | \(=\) | \(136\) |
| \(2x\) | \(=\) | \(134\) |
| \(x\) | \(=\) | \(67\) |
The numbers are \(67\) and \(69\).
Let \(x\) be the smaller angle in degrees, so the bigger one is \(x + 30\). Angles on a straight line add to \(180^\circ\).
| \(x + (x + 30)\) | \(=\) | \(180\) |
| \(2x + 30\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(150\) |
| \(x\) | \(=\) | \(75\) |
The angles are \(75^\circ\) and \(105^\circ\). Check: \(75 + 105 = 180\).
Let \(x\) be Tom’s age now, so his father is \(3x\). In \(15\) years they will be \(x + 15\) and \(3x + 15\).
| \(2(x + 15)\) | \(=\) | \(3x + 15\) |
| \(2x + 30\) | \(=\) | \(3x + 15\) |
| \(15\) | \(=\) | \(x\) |
Tom is \(15\) and his father is \(45\). Check: in \(15\) years they are \(30\) and \(60\), and \(60 = 2 \times 30\).
| \(45.50\) | \(=\) | \(29 + 0.15m\) |
| \(16.50\) | \(=\) | \(0.15m\) |
| \(m\) | \(=\) | \(\dfrac{16.50}{0.15} = 110\) |
\(110\) extra minutes were used. Check: \(29 + 0.15 \times 110 = 45.50\).
(i) Write an equation, with Ana’s share \(\$x\).
(ii) Find each share.
| \(x + (x + 10) + 3x\) | \(=\) | \(120\) |
| \(5x + 10\) | \(=\) | \(120\) |
| \(5x\) | \(=\) | \(110\) |
| \(x\) | \(=\) | \(22\) |
(i) \(x + (x + 10) + 3x = 120\)
(ii) Ana \(\$22\), Ben \(\$32\), Cy \(\$66\). Check: \(22 + 32 + 66 = 120\).
Let \(t\) be the number of hours from now. Ben’s distance \(=\) Sara’s distance.
| \(24 + 14t\) | \(=\) | \(26t\) |
| \(24\) | \(=\) | \(12t\) |
| \(t\) | \(=\) | \(2\) |
Sara catches Ben after \(2\) hours, when both are \(52\) km along the track.
Common pitfalls
Frequently asked questions
How do I turn a word problem into an equation?
Let a pronumeral stand for the unknown, write every other quantity using it, and use the words is, gives or altogether as the equals sign.
How do I write consecutive numbers?
\(x\) and \(x + 1\); consecutive even or odd numbers are \(x\) and \(x + 2\).
How do I set up an age problem?
Write each age now with the pronumeral, then add the same number of years to every age for the future.
Why answer in words?
The question asks about the story, so give the value with its units, such as 75 degrees or 2 hours.
How do I check a word problem?
Put your answer back into the story and see that every condition is true.