Writing an Expression

What you'll learn

How to use letters to stand for unknown or changing numbers.
How to turn words like “more than”, “half”, “each” and “total” into algebra.
How to write expressions for costs, ages, times and quantities.
How to tell the difference between an expression and a formula.

1. The building blocks

In algebra, we often do not know the exact number yet. Instead, we use a letter such as $x$ , $n$ or $m$ .

Definition

Variable and expression

A variable is a letter that stands for a number that may change or is unknown, such as $x$ or $n$ .
An expression is a combination of numbers, variables and operations, such as $x + 6$ or $5 m$ . An expression does not need an equals sign.

For example, if an adult ticket costs $x$ pounds, then $x$ is the variable. If a child ticket costs half as much, we write an expression using $x$ .

Example

Writing half of a price

A swimming pool adult ticket costs $x$ pounds. A junior ticket costs half the adult price. Write an expression for the junior ticket price.

Start with the adult price: $x$ pounds.
The word half means divide by 2.
So the junior ticket price, in pounds, is:
$x2\frac{x}{2}$

Key Idea

What an expression does

An expression describes a value without needing to know the exact number yet. If you later find out that $x = 10$ , then $x2\frac{x}{2}$ would be 5.

2. Choosing the correct operation

Most “writing an expression” questions are about translating words into operations.

Useful word clues:

more than, older than, longer than usually mean add.
less than, younger than, eats, spends, loses usually mean subtract.
each, per, for every, twice, three times usually mean multiply.
shared equally, half, or “how many can be bought from a total” usually mean divide.

Example

More than and less than

Ella has $m$ marbles. Tom has 7 more marbles than Ella. Sam has 4 fewer marbles than Ella. Write expressions for Tom and Sam.

Ella has $m$ marbles.
Tom has 7 more, so add 7:
$m + 7$
Sam has 4 fewer, so subtract 4:
$m - 4$

Common Mistake

Mixing up subtraction order

“4 fewer than $m$ ” means start with $m$ and take away 4, so it is $m - 4$ . It is not $4 - m$ .

3. Multiplication in algebra

When you have repeated equal amounts, you multiply.

For example, if one hour of work pays $h$ pounds, then 10 hours pays $10 h$ pounds.

Definition

Coefficient

A coefficient is the number multiplying a variable. In $10 h$ , the coefficient is 10, meaning 10 lots of $h$ .

In algebra, we usually write multiplication without the multiplication sign:

$\times x$ is written as $5 x$ .
$\times m$ is written as $3 m$ .
$\times m$ is written as $bm$ .

Example

Pay per hour

A tutor earns $p$ pounds per hour. One week, the tutor works for 12 hours. Write an expression for the total pay that week.

One hour earns $p$ pounds.
12 hours means 12 lots of $p$ .
Multiply 12 by $p$ :
$12 p$

Common Mistake

Adding instead of multiplying

If something happens “for each” item or “per” hour, you usually multiply. For 12 hours at $p$ pounds per hour, write $12 p$ , not $p + 12$ .

4. Totals with two different items

Sometimes there are two types of item, each with its own cost or amount. Work out each part separately, then add them.

Example

Cost of two types of item

A bottle of juice costs $j$ pounds. A sandwich costs $s$ pounds. Write an expression for the cost of 4 bottles of juice and 3 sandwiches.

One bottle of juice costs $j$ pounds, so 4 bottles cost $4 j$ pounds.
One sandwich costs $s$ pounds, so 3 sandwiches cost $3 s$ pounds.
The word and tells us to add the two costs.
The total cost, in pounds, is:
$4 j + 3 s$

Tip

Build totals in chunks

For total cost or total number of objects, write one small expression for each type first, then add the chunks together.

5. Division: working backwards from a total

Division often appears when you know the total amount and the amount for one item.

For example, if each ticket costs £5 and the total money collected is $x$ pounds, then the number of tickets is the total divided by 5.

Example

Finding how many tickets were sold

Tickets for a concert cost £4 each. The total money collected is $y$ pounds. Write an expression for the number of tickets sold.

Each ticket costs £4.
The total money collected is $y$ pounds.
To find how many tickets were sold, divide the total money by the price of one ticket.
The number of tickets sold is:
$y4\frac{y}{4}$

Tip

Fraction bar means divide

The expression $y4\frac{y}{4}$ means $y$ divided by 4. Fractions are often the neatest way to write division in algebra.

6. Building related expressions

Some questions give one person or object first, then describe others compared with it. Write each expression carefully before finding any total.

This is common with ages.

Example

Ages and a total

Lina is $n$ years old. Her brother is 6 years older than Lina. Her cousin is 3 years younger than Lina. Write an expression for the total of their three ages.

Lina’s age is $n$ .
Her brother is 6 years older, so his age is $n + 6$ .
Her cousin is 3 years younger, so the cousin’s age is $n - 3$ .
Add the three ages:
$n + (n + 6) + (n - 3)$
Collect the $n$ terms and the number terms:
$n+(n+6)+(n−3)=n+n+n+6−3=3n+3\begin{aligned} n+(n+6)+(n-3) &= n+n+n+6-3 \\ &= 3n+3 \end{aligned}$

Definition

Like terms

Like terms have the same variable part, so they can be collected together. For example, $n + n + n = 3 n$ , but $n$ and 6 are not like terms.

7. Expressions for packs and contents

Be careful when a question asks for the number of packs compared with the number of items inside the packs.

If a shop has $s$ small packs and $b$ big packs:

total packs means just add the packs: $s + b$ .
total items means multiply by how many items are in each pack, then add.

Example

Packs compared with items inside

A stationery shop has $s$ small boxes of pencils and $l$ large boxes of pencils. A small box contains 6 pencils. A large box contains 10 pencils.

Write an expression for the total number of boxes, and then for the total number of pencils.

The total number of boxes is the number of small boxes plus the number of large boxes:
$s + l$
Each small box contains 6 pencils, so the small boxes contain $6 s$ pencils.
Each large box contains 10 pencils, so the large boxes contain $10 l$ pencils.
The total number of pencils is:
$6 s + 10 l$

Common Mistake

Answering the wrong total

If the question asks for total packs, do not multiply by the number inside each pack. Only multiply when the question asks for total items inside the packs.

8. Expressions versus formulae

Most of the time, you are asked to “write an expression”. That means you usually do not need an equals sign.

A formula links variables using an equals sign. It often tells you how to calculate one variable from others.

Definition

Formula

A formula is an equation that shows a relationship between variables, such as $C = 40 m + 65 j$ .

Example

Writing a formula for cost

Muffins cost 40p each. Juice cartons cost 65p each. The total cost of $m$ muffins and $j$ juice cartons is $C$ pence. Write a formula for $C$ .

The cost of $m$ muffins is $40 m$ pence.
The cost of $j$ juice cartons is $65 j$ pence.
Add the two costs together.
Because the total cost is called $C$ , write:
$C = 40 m + 65 j$

Tip

Expression or formula?

If the question says “write an expression”, give the algebra part, such as $40 m + 65 j$ . If it says “write a formula for $C$ ”, include $C =$ .

A quick translation checklist

When you read a word problem, ask yourself:

What does each letter stand for?
Am I finding a price, age, time, number of items, or total?
Which operation is needed: add, subtract, multiply or divide?
Do I need an expression, or a formula with an equals sign?

Exam technique

In the exam

Underline key words such as “more than”, “each”, “half”, “total” and “younger than”.
Write one small expression for each part before combining them.
Check the units: pounds, pence, minutes, tickets, packs and items can lead to different expressions.

Self review

Check yourself

If one item costs $x$ pounds, what expression represents 8 of those items?
What is the difference between $b + 5$ and $5 b$ in words?
When would you write $C = 25 a + 30 b$ instead of just $25 a + 30 b$ ?

Writing an Expression

What you'll learn

How to use letters to stand for unknown or changing numbers.
How to turn words like “more than”, “half”, “each” and “total” into algebra.
How to write expressions for costs, ages, times and quantities.
How to tell the difference between an expression and a formula.

1. The building blocks

In algebra, we often do not know the exact number yet. Instead, we use a letter such as $x$ , $n$ or $m$ .

Definition

Variable and expression

A variable is a letter that stands for a number that may change or is unknown, such as $x$ or $n$ .
An expression is a combination of numbers, variables and operations, such as $x + 6$ or $5 m$ . An expression does not need an equals sign.

For example, if an adult ticket costs $x$ pounds, then $x$ is the variable. If a child ticket costs half as much, we write an expression using $x$ .

Example

Writing half of a price

A swimming pool adult ticket costs $x$ pounds. A junior ticket costs half the adult price. Write an expression for the junior ticket price.

Start with the adult price: $x$ pounds.
The word half means divide by 2.
So the junior ticket price, in pounds, is:
$x2\frac{x}{2}$

Key Idea

What an expression does

An expression describes a value without needing to know the exact number yet. If you later find out that $x = 10$ , then $x2\frac{x}{2}$ would be 5.

2. Choosing the correct operation

Most “writing an expression” questions are about translating words into operations.

Useful word clues:

more than, older than, longer than usually mean add.
less than, younger than, eats, spends, loses usually mean subtract.
each, per, for every, twice, three times usually mean multiply.
shared equally, half, or “how many can be bought from a total” usually mean divide.

Example

More than and less than

Ella has $m$ marbles. Tom has 7 more marbles than Ella. Sam has 4 fewer marbles than Ella. Write expressions for Tom and Sam.

Ella has $m$ marbles.
Tom has 7 more, so add 7:
$m + 7$
Sam has 4 fewer, so subtract 4:
$m - 4$

Common Mistake

Mixing up subtraction order

“4 fewer than $m$ ” means start with $m$ and take away 4, so it is $m - 4$ . It is not $4 - m$ .

3. Multiplication in algebra

When you have repeated equal amounts, you multiply.

For example, if one hour of work pays $h$ pounds, then 10 hours pays $10 h$ pounds.

Definition

Coefficient

A coefficient is the number multiplying a variable. In $10 h$ , the coefficient is 10, meaning 10 lots of $h$ .

In algebra, we usually write multiplication without the multiplication sign:

$\times x$ is written as $5 x$ .
$\times m$ is written as $3 m$ .
$\times m$ is written as $bm$ .

Example

Pay per hour

A tutor earns $p$ pounds per hour. One week, the tutor works for 12 hours. Write an expression for the total pay that week.

One hour earns $p$ pounds.
12 hours means 12 lots of $p$ .
Multiply 12 by $p$ :
$12 p$

Common Mistake

Adding instead of multiplying

If something happens “for each” item or “per” hour, you usually multiply. For 12 hours at $p$ pounds per hour, write $12 p$ , not $p + 12$ .

4. Totals with two different items

Sometimes there are two types of item, each with its own cost or amount. Work out each part separately, then add them.

Example

Cost of two types of item

A bottle of juice costs $j$ pounds. A sandwich costs $s$ pounds. Write an expression for the cost of 4 bottles of juice and 3 sandwiches.

One bottle of juice costs $j$ pounds, so 4 bottles cost $4 j$ pounds.
One sandwich costs $s$ pounds, so 3 sandwiches cost $3 s$ pounds.
The word and tells us to add the two costs.
The total cost, in pounds, is:
$4 j + 3 s$

Tip

Build totals in chunks

For total cost or total number of objects, write one small expression for each type first, then add the chunks together.

5. Division: working backwards from a total

Division often appears when you know the total amount and the amount for one item.

For example, if each ticket costs £5 and the total money collected is $x$ pounds, then the number of tickets is the total divided by 5.

Example

Finding how many tickets were sold

Tickets for a concert cost £4 each. The total money collected is $y$ pounds. Write an expression for the number of tickets sold.

Each ticket costs £4.
The total money collected is $y$ pounds.
To find how many tickets were sold, divide the total money by the price of one ticket.
The number of tickets sold is:
$y4\frac{y}{4}$

Tip

Fraction bar means divide

The expression $y4\frac{y}{4}$ means $y$ divided by 4. Fractions are often the neatest way to write division in algebra.

6. Building related expressions

Some questions give one person or object first, then describe others compared with it. Write each expression carefully before finding any total.

This is common with ages.

Example

Ages and a total

Lina is $n$ years old. Her brother is 6 years older than Lina. Her cousin is 3 years younger than Lina. Write an expression for the total of their three ages.

Lina’s age is $n$ .
Her brother is 6 years older, so his age is $n + 6$ .
Her cousin is 3 years younger, so the cousin’s age is $n - 3$ .
Add the three ages:
$n + (n + 6) + (n - 3)$
Collect the $n$ terms and the number terms:
$n+(n+6)+(n−3)=n+n+n+6−3=3n+3\begin{aligned} n+(n+6)+(n-3) &= n+n+n+6-3 \\ &= 3n+3 \end{aligned}$

Definition

Like terms

Like terms have the same variable part, so they can be collected together. For example, $n + n + n = 3 n$ , but $n$ and 6 are not like terms.

7. Expressions for packs and contents

Be careful when a question asks for the number of packs compared with the number of items inside the packs.

If a shop has $s$ small packs and $b$ big packs:

total packs means just add the packs: $s + b$ .
total items means multiply by how many items are in each pack, then add.

Example

Packs compared with items inside

A stationery shop has $s$ small boxes of pencils and $l$ large boxes of pencils. A small box contains 6 pencils. A large box contains 10 pencils.

Write an expression for the total number of boxes, and then for the total number of pencils.

The total number of boxes is the number of small boxes plus the number of large boxes:
$s + l$
Each small box contains 6 pencils, so the small boxes contain $6 s$ pencils.
Each large box contains 10 pencils, so the large boxes contain $10 l$ pencils.
The total number of pencils is:
$6 s + 10 l$

Common Mistake

Answering the wrong total

If the question asks for total packs, do not multiply by the number inside each pack. Only multiply when the question asks for total items inside the packs.

8. Expressions versus formulae

Most of the time, you are asked to “write an expression”. That means you usually do not need an equals sign.

A formula links variables using an equals sign. It often tells you how to calculate one variable from others.

Definition

Formula

A formula is an equation that shows a relationship between variables, such as $C = 40 m + 65 j$ .

Example

Writing a formula for cost

Muffins cost 40p each. Juice cartons cost 65p each. The total cost of $m$ muffins and $j$ juice cartons is $C$ pence. Write a formula for $C$ .

The cost of $m$ muffins is $40 m$ pence.
The cost of $j$ juice cartons is $65 j$ pence.
Add the two costs together.
Because the total cost is called $C$ , write:
$C = 40 m + 65 j$

Tip

Expression or formula?

If the question says “write an expression”, give the algebra part, such as $40 m + 65 j$ . If it says “write a formula for $C$ ”, include $C =$ .

A quick translation checklist

When you read a word problem, ask yourself:

What does each letter stand for?
Am I finding a price, age, time, number of items, or total?
Which operation is needed: add, subtract, multiply or divide?
Do I need an expression, or a formula with an equals sign?

Exam technique

In the exam

Underline key words such as “more than”, “each”, “half”, “total” and “younger than”.
Write one small expression for each part before combining them.
Check the units: pounds, pence, minutes, tickets, packs and items can lead to different expressions.

Self review

Check yourself

If one item costs $x$ pounds, what expression represents 8 of those items?
What is the difference between $b + 5$ and $5 b$ in words?
When would you write $C = 25 a + 30 b$ instead of just $25 a + 30 b$ ?

Writing an Expression

What you'll learn

1. The building blocks

2. Choosing the correct operation

3. Multiplication in algebra

4. Totals with two different items

5. Division: working backwards from a total

6. Building related expressions

7. Expressions for packs and contents

8. Expressions versus formulae

A quick translation checklist

Recap questions

You've reached the end

Writing an Expression

What you'll learn

1. The building blocks

2. Choosing the correct operation

3. Multiplication in algebra

4. Totals with two different items

5. Division: working backwards from a total

6. Building related expressions

7. Expressions for packs and contents

8. Expressions versus formulae

A quick translation checklist

Recap questions

You've reached the end

Writing an Expression

What you'll learn

1. The building blocks

2. Choosing the correct operation

3. Multiplication in algebra

4. Totals with two different items

5. Division: working backwards from a total

6. Building related expressions

7. Expressions for packs and contents

8. Expressions versus formulae

A quick translation checklist

Recap questions

You've reached the end

Writing an Expression Revision Guide

Writing an Expression

What you'll learn

1. The building blocks

2. Choosing the correct operation

3. Multiplication in algebra

4. Totals with two different items

5. Division: working backwards from a total

6. Building related expressions

7. Expressions for packs and contents

8. Expressions versus formulae

A quick translation checklist

Recap questions

You've reached the end

Writing an Expression Revision Guide