Words to Algebraic Expressions
Every question with a fully worked solution.
- Words to Algebraic Expressions - Video - Translating words into algebraic expressions Watch
Theory
To turn words into algebra, swap each key word for its operation and each unknown number for a pronumeral. This Year 7 maths page lists the key words, shows when the order swaps (“\(5\) less than \(x\)” is \(x - 5\)), how grouping words like “half of the sum” work, and how to write expressions for money and shapes.
Four key words appear again and again: the sum of \(x\) and \(y\) is \(x + y\); the difference between \(x\) and \(y\) is \(x - y\); the product of \(x\) and \(y\) is \(xy\); the quotient when \(p\) is divided by \(q\) is \(\dfrac{p}{q}\).
Write the answer in concise notation: \(3 \times m = 3m\), \(a \times a = a^2\), and a division as a fraction.
Order matters for subtraction and division. “The difference between \(a\) and \(b\)” keeps the order of the words: \(a - b\). But “\(5\) less than \(x\)” swaps it: start with \(x\) and take \(5\) away, so it is \(x - 5\).
Grouping words. “Half of the sum” or “the cube of the difference” act on a whole group. Build the group first, then apply the outer operation to all of it. A square root sign covers the whole group too: the square root of the difference between \(m\) and \(n\) is \(\sqrt{m - n}\).
Check your expression by putting in friendly numbers.
Key words:
| Operation | Key words |
|---|---|
| add \((+)\) | sum, plus, more than, increased by, total |
| subtract \((-)\) | difference, minus, less than, decreased by, fewer |
| multiply \((\times)\) | product, times, lots of, twice, double |
| divide \((\div)\) | quotient, divided by, half of, one third of, shared equally |
| power, root | square of \((a^2)\), cube of \((a^3)\), square root of \((\sqrt{a})\) |
How to write words as an expression
- Choose a pronumeral for each unknown number.
- Swap each key word for its operation.
- Build any group first, then apply the outer operation to all of it.
- Write in concise notation and check with friendly numbers.
(i) the product of \(6\) and \(y\)
(ii) \(k\) decreased by \(4\)
(iii) the quotient when \(m\) is divided by \(8\)
(iv) \(9\) less than \(t\)
Product means multiply; decreased by means subtract; quotient means divide (write a fraction). For the last one, start with \(t\), then take \(9\) away. Check with \(t = 20\): \(20 - 9 = 11\).
(i) \(6y\)
(ii) \(k - 4\)
(iii) \(\dfrac{m}{8}\)
(iv) \(t - 9\)
(i) \(2x + 7\)
(ii) \(4(n - 3)\)
Read one operation at a time. Check the first with \(x = 5\): \(2 \times 5 + 7 = 17\), and \(7\) more than \(10\) is \(17\). Check the second with \(n = 5\): \(4(5 - 3) = 8\).
(i) \(7\) more than twice \(x\)
(ii) \(4\) times the difference between \(n\) and \(3\)
The pies cost \(\$5p\). Take this away from \(\$50\). Check with \(p = 4\): \(4\) pies cost \(\$20\) and \(50 - 20 = 30\); also \(50 - 5 \times 4 = 30\).
Mia has \(\$(50 - 5p)\) left.
(i) the square of the sum of \(x\) and \(y\)
(ii) the square root of the product of \(4\) and \(x\)
(iii) the square root of the difference between the square of \(m\) and \(9\)
Build the group first, then apply the outer operation to all of it. Check (i) with \(x = 2\), \(y = 3\): \((2 + 3)^2 = 25\), but \(x + y^2 = 11\), so the brackets matter.
(i) \((x + y)^2\)
(ii) \(\sqrt{4x}\)
(iii) \(\sqrt{m^2 - 9}\)
\(n \times n \times n \times n = n^4\). Then subtract \(1\).
\(n^4 - 1\)
(ii) A square has perimeter \(3y + 2\). Write the length of one side.
(iii) A parallelogram has base \(b\) and height \(h\). Write its area.
(i) Add all four sides: \(p + p + q + q\).
(ii) A square has \(4\) equal sides, so divide the whole perimeter by \(4\).
(iii) Area of a parallelogram is base times height.
(i) \(2p + 2q\)
(ii) \(\dfrac{3y + 2}{4}\)
(iii) \(bh\)
Common pitfalls
Frequently asked questions
What does sum mean in algebra?
Add. The sum of \(x\) and \(y\) is \(x + y\).
What does product mean?
Multiply. The product of \(m\) and \(n\) is \(mn\).
What does quotient mean?
Divide. The quotient when \(p\) is divided by \(q\) is \(\dfrac{p}{q}\).
How do you write 5 less than x?
\(x - 5\): start with \(x\) and take \(5\) away.
How do you write half of the sum of m and 6?
\(\dfrac{m + 6}{2}\): build the sum, then halve all of it.