# Turning Words into Algebra

> Write expressions and equations from word problems about prices, ages, perimeters and shares, then solve them.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/algebra-word-problems/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Uncle Sam's riddle
At a family barbecue, Jack's uncle gives him a riddle. "I am 15 years older than you. In 3 years' time, our ages will add up to 53. How old am I?"

Jack grabs a napkin. He does not know his uncle's age, so he calls it $u$. Then he is $u - 15$. In 3 years they will be $u + 3$ and $u - 15 + 3$.

He writes $u + 3 + u - 12 = 53$, which is $2u - 9 = 53$. So $2u = 62$ and $u = 31$. "You are 31!" His uncle laughs. Jack turned a puzzle in words into algebra, and the algebra did the rest.

**In short:** To turn a word problem into algebra, choose a letter for the unknown, write the other amounts using it, and use the facts to make an equation. Sum means add, difference means subtract, product means multiply and quotient means divide.

## Before you start

- Collecting like terms and expanding brackets (earlier chapters in this unit).
- Solving equations such as $2u - 9 = 53$ (see Solving Equations).
- **Perimeter** is the distance around a shape. **Average** is the total divided by how many there are.

## Word to symbol dictionary

| Words | Algebra |
|---|---|
| sum of $a$ and $b$, $a$ plus $b$, $b$ more than $a$ | $a + b$ |
| difference, $b$ less than $a$, $a$ take away $b$ | $a - b$ |
| product of $a$ and $b$, $a$ times $b$ | $ab$ |
| quotient, $a$ divided by $b$, $a$ shared among $b$ | $\frac{a}{b}$ |
| double $a$, twice $a$ | $2a$ |
| half of $a$ | $\frac{a}{2}$ or $\frac{1}{2}a$ |
| 3 times as many as $a$ | $3a$ |
| the square of $a$ | $a^2$ |

### Watch out: "Less than" flips the order
The words 5 less than $n$ mean $n - 5$, not $5 - n$. You start with $n$ and take 5 away.

## Writing expressions

### Worked example: Shopping
A mango costs $z$ dollars. Three mangoes cost $3z$ dollars.

Three watermelons cost $3x + 9$ dollars. One watermelon costs $\frac{3x + 9}{3} = x + 3$ dollars.

A chair costs $x$ dollars. With 100 dollars you can buy $\frac{100}{x}$ chairs.

### Worked example: Ages
Raymond is $y$ years old. In 12 years he will be $y + 12$.

A woman is 3 times as old as her son, who is $x$. She is $3x$ now, and in 1 year she will be $3x + 1$.

### Worked example: Shapes and averages
A rectangle is 7 cm long and $x$ cm wide. Perimeter $= 7 + x + 7 + x = 2x + 14$ cm.

Lisa weighs 38 kg and Gina weighs $K$ kg. Their average weight is $\frac{38 + K}{2}$ kg.

## Writing and solving equations

1. Choose a letter for the thing you do not know. Say what it means.
2. Write the other amounts using that letter.
3. Use the fact in the question to make an equation.
4. Solve it, then answer the question in words.

### Worked example: Half its own value
When $Z$ is added to half of itself, the result is 39.

$Z + \frac{1}{2}Z = 39$, so $\frac{3}{2}Z = 39$, which gives $Z = 26$.

### Worked example: Ages again
A mother is 37 and her children are 12 and 7. In how many years will the children's ages add up to the mother's age?

Let $n$ be the years. Children: $12 + n$ and $7 + n$. Mother: $37 + n$.

$19 + 2n = 37 + n$, so $n = 18$. In 18 years: children 30 and 25 (total 55), mother 55.

### Exam tip: Each person ages too
In age problems, **everyone** gets older by the same number of years. Do not add the years to only one person.

### Challenge: Selective level: substitute after writing
Jason's mother buys $2x$ kg of chicken at 3.80 dollars per kg, $(1 + x)$ kg of beef at 7.50 dollars per kg and $4\frac{1}{2}$ litres of oil at 5.40 dollars per litre. If $x = 2$, how much does she spend?

Chicken: $4 \times 3.80 = 15.20$. Beef: $3 \times 7.50 = 22.50$. Oil: $4.5 \times 5.40 = 24.30$.

Total: 62.00 dollars.

Ratio with a letter: $60k$ marbles are shared 2 : 3. There are 5 parts, so one part is $12k$. Alan gets $24k$ and George gets $36k$. If Alan has 192 marbles, then $24k = 192$ and $k = 8$.

**Mind map**

```text
Words into algebra
  Dictionary
    sum, more than: +
    difference, less than: -
    product, times: x
    quotient, shared: ÷
  Expressions
    Prices: $3z$
    Ages: $y + 12$
    Perimeter: $2x + 14$
  Equations
    Name the unknown
    Use the fact to write =
    Solve and answer in words
  Traps
    5 less than $n$ is $n - 5$
    Everyone ages together
```

## Quick summary

- Choose a letter for the unknown and say what it means.
- Learn the key words: sum means add, difference means subtract, product means multiply and quotient means divide.
- 5 less than $n$ means $n - 5$.
- Write other amounts using the letter, then use the fact in the question to make an equation.
- In age problems, everybody gets older by the same amount.
- Finish by answering the actual question in words.

## Quiz

Try these questions from the chapter quiz.

1. A pen costs $p$ dollars. What do 6 pens cost?
   - A. $p + 6$
   - B. $6p$
   - C. $\frac{p}{6}$
   - D. $6 + p$ dollars
   - Answer: B. $6p$. 6 pens cost 6 lots of $p$, which is $6p$ dollars.

2. Leo is $z$ cm taller than Bob, who is 143 cm tall. How tall is Leo?
   - A. $143 - z$
   - B. $143 + z$
   - C. $143z$
   - D. $z - 143$
   - Answer: B. $143 + z$. Taller means more, so add: $143 + z$ cm.

3. How do you write 7 less than $m$ in algebra?
   - A. $7 - m$
   - B. $7m$
   - C. $m - 7$
   - D. $m + 7$
   - Answer: C. $m - 7$. Start with $m$ and take 7 away.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/algebra-word-problems/#quiz