Using Letters to Represent Numbers

Why algebra uses letters, what a letter stands for, how to write a simple algebraic expression from words, and how to name the terms, coefficients and constant term of an expression.

Before you start

In Primary, a box stood for a missing number. In $\square - 5 = 2$ the box is 7, because $7 - 5 = 2$ .

A division can be written as a fraction: $3 \div 4 = \frac{3}{4}$ .

Subtracting a number is the same as adding its negative: $10 - 3 = 10 + (-3) = 7$ .

Why Letters Are Used

In the puzzle $\square - 5 = 2$ , the box keeps the place of a number we do not know yet. In algebra a letter keeps that place instead of a box: $y - 5 = 2$ is the same puzzle, and here $y$ stands for 7. Finding such numbers is the work of a later chapter, on equations.

A letter has a second, more important use. It can stand for any number, so that one line of algebra describes every case at once. The example below works two cases with numbers first, and then writes the rule for every case with a letter.

Example 1: Sharing money among 3 children

Mr Tan shares a sum of money equally among his 3 grandchildren. How much does each child get if the sum is
(a) $270,
(b) $3300,
(c) $z$ dollars?

Solution

(a) Sharing equally among 3 means dividing by 3: $270 \div 3 = 90$ . Each child gets $90.

(b) In the same way, $3300 \div 3 = 1100$ . Each child gets $1100.

(c) The working does not change when the sum changes: whatever the sum is, it is divided by 3. A sum of $z$ dollars gives each child $z \div 3$ dollars.

Written as a fraction, $z \div 3$ is $\frac{z}{3}$ .

(a) $90 (b) $1100 (c) Each child gets $\frac{z}{3}$ dollars. This one line holds for every sum of money: $z \div 3$ is the rule, and (a) and (b) are two cases of it.

Definition: variable

In algebra we use a letter to represent a number. The letter may stand for a number we do not know yet, or for any number. A letter used in this way is called a variable.

Any letter can be used ( $a$ , $b$ , $k$ , $n$ , $p$ , $x$ , $y$ , $z$ , …). When there is more than one quantity, we use a different letter for each one.

Where the word "algebra" comes from

The word algebra comes from the Arabic al-jabr, meaning "restoration" or "reunion of broken parts". It is taken from the title of a book written about the year 820 by the mathematician al-Khwarizmi, who worked in Baghdad.

A Letter Stands for a Number

A letter in algebra always stands for a number: a number of things, an amount of money, a length, a speed. It never stands for the things themselves.

If Ali buys $a$ apples, then $a$ stands for the number of apples, such as 4 or 10. It does not mean "apple". If each apple costs 60 cents, Ali pays $60 \times a$ cents, written $60a$ cents, and $60a$ is a number of cents too.

When a letter stands for a measurement, the unit is written outside it. "The length is $x$ cm" means that $x$ is the number of centimetres, such as 12 or 7.5.

Example 2: Saying what a letter and an expression stand for

(a) There are $n$ students in a class. Five more students join the class. What does $n + 5$ stand for?
(b) A car travels at $v$ km/h for 3 hours. What does $3v$ stand for?

Solution

(a) $n$ is the number of students in the class at first. When 5 more join, the number of students becomes 5 more than $n$ , which is $n + 5$ .

So $n + 5$ is a number: the number of students after the 5 join. If there were 35 students at first, there are now 40.

(b) $v$ is the speed in km/h, a number such as 60 or 80. Distance = speed × time, so in 3 hours the car travels $v \times 3$ km.

A number times a letter is written with the number first and no × sign: $v \times 3$ is written $3v$ .

(a) $n + 5$ stands for the number of students in the class after the 5 students join. (b) $3v$ stands for the distance, in kilometres, that the car travels in 3 hours.

Algebraic Expressions

In Example 1 the share of each child was $z \div 3$ dollars, and in Example 2 the number of students was $n + 5$ . Each of $z \div 3$ and $n + 5$ is made of numbers and letters joined by the signs of arithmetic. Such a thing has a name.

Definition: algebraic expression

An algebraic expression is made of numbers and variables joined by the operation signs $+$ , $-$ , $\times$ and $\div$ . It may also contain brackets.

Examples: $n + 5$ , $y - 5$ , $3v$ , $\frac{z}{3}$ , $2m + 1$ , $x + 2y - 3z + 4$ , $\frac{k + 4}{2}$ .

An expression is written in terms of the letters in it: $\frac{z}{3}$ is an expression in terms of $z$ ; $x + 2y - 3z + 4$ is an expression in terms of $x$ , $y$ and $z$ .

An expression has no equals sign. $y - 5$ is an expression; $y - 5 = 2$ is a statement that two things are equal, called an equation, which is the subject of a later chapter.

To write an expression from words, say what you would do to a number, and then do the same to the letter. Some phrases and their expressions:

In words	Working	Expression
5 more than $n$	add 5 to $n$	$n + 5$
$n$ minus 3, or subtract 3 from $n$	take 3 away from $n$	$n - 3$
the sum of $p$ and $q$	add $p$ and $q$	$p + q$
4 times $k$	$4 \times k$	$4k$
twice $m$	$2 \times m$	$2m$
$k$ shared equally among 6	$k \div 6$	$\frac{k}{6}$
half of $b$	$b \div 2$	$\frac{b}{2}$

How the short forms are written

$4 \times k$ is written $4k$ : the × sign is left out and the number comes first, never $k4$ . A division is written as a fraction: $k \div 6$ is written $\frac{k}{6}$ . The lesson on algebraic notation explains these ways of writing, and the others algebra uses, in full.

Example 3: Numbers first, then the letter

A packet holds 8 biscuits. How many biscuits are there in
(a) 5 packets,
(b) 12 packets,
(c) $k$ packets?

Solution

(a) Each packet holds 8 biscuits, so 5 packets hold $8 \times 5 = 40$ biscuits.

(b) In the same way, 12 packets hold $8 \times 12 = 96$ biscuits.

(a) 40 biscuits (b) 96 biscuits (c) There are $8k$ biscuits in $k$ packets.

Example 4: Writing expressions from words

Write an algebraic expression for each of these.
(a) 7 more than $x$
(b) Subtract 10 from $y$
(c) The product of 9 and $t$
(d) $w$ divided by 5
(e) 1 more than twice $m$

Solution

(a) "7 more than $x$ " means add 7 to $x$ : $x + 7$ .

(b) "Subtract 10 from $y$ " means start with $y$ and take 10 away: $y - 10$ . The number you subtract from comes first.

(d) $w$ divided by 5 is $w \div 5$ , written $\frac{w}{5}$ .

(e) Twice $m$ is $2 \times m = 2m$ . One more than that is $2m + 1$ .

(a) $x + 7$ (b) $y - 10$ (c) $9t$ (d) $\frac{w}{5}$ (e) $2m + 1$

The Same Letter Means the Same Number

Inside one expression, a letter stands for one number all the way through. In $n + n$ , if the first $n$ is 6, then the second $n$ is 6 as well, and $n + n$ is $6 + 6 = 12$ .

Two different letters stand for numbers that may be different, but they do not have to be. In $p + q$ , $p$ could be 3 and $q$ could be 8; or $p$ and $q$ could both be 3.

Rule: one letter, one number

In one expression, the same letter stands for the same number every time it appears.

Different letters may stand for different numbers or for the same number.

Example 5: Which choices are allowed?

Say whether each choice of numbers is allowed.
(a) In $m + m + 4$ , the first $m$ is 3 and the second $m$ is 5.
(b) In $m + n$ , $m$ is 3 and $n$ is 3.
(c) In $m + n$ , $m$ is 3 and $n$ is 8.

Solution

(a) Both $m$ s are in the same expression, so they must stand for the same number. One cannot be 3 while the other is 5.

(b) $m$ and $n$ are different letters. Different letters may stand for the same number, so both may be 3.

(a) Not allowed (b) Allowed (c) Allowed

Terms, Coefficients and Constant Terms

An expression is made of parts called terms, joined by $+$ and $-$ signs. Since subtracting is the same as adding the negative,

$x + 2y - 3z + 4 = x + 2y + (-3z) + 4$ ,

so the four terms of $x + 2y - 3z + 4$ are $x$ , $2y$ , $-3z$ and $4$ . A term with no sign in front of it, at the start of an expression, is positive.

Definitions: term, coefficient, constant term

The terms of an expression are its parts joined by $+$ and $-$ . Each term takes the sign in front of it. A bracket, or a fraction with a sum above or below its line, counts as one part: the signs inside it do not split it. So $\frac{k + 4}{2}$ is one term, not two.

In a term with letters, the number that multiplies the letters, together with its sign, is the coefficient. In $2y$ the coefficient of $y$ is 2; in $-3z$ the coefficient of $z$ is $-3$ ; in $-6ab$ the coefficient of $ab$ is $-6$ .

A letter on its own has coefficient 1, because $x = 1 \times x$ ; the 1 is not written. In the same way, $-x$ has coefficient $-1$ .

A term with no letter is the constant term, with its sign. The constant term of $x + 2y - 3z + 4$ is 4; the constant term of $5 - 4p$ is 5, and the coefficient of $p$ there is $-4$ .

Example 6: The terms of x + 2y − 3z + 4

For the expression $x + 2y - 3z + 4$ , write down
(a) its terms,
(b) the coefficients of $x$ , $y$ and $z$ ,
(c) its constant term.

Solution

(a) Split the expression at each $+$ and $-$ sign, keeping each sign with the term after it: $x$ , $+2y$ , $-3z$ , $+4$ .

(b) $x$ is $1 \times x$ , so the coefficient of $x$ is 1. The coefficient of $y$ is 2. The term $-3z$ is $-3 \times z$ , so the coefficient of $z$ is $-3$ , with its minus sign.

(a) The terms are $x$ , $2y$ , $-3z$ and $4$ . (b) The coefficient of $x$ is 1, of $y$ is 2, and of $z$ is $-3$ . (c) The constant term is 4.

A term may be a number multiplying more than one letter, or a letter multiplied by itself. Here $ab$ means $a \times b$ , $a^{2}$ means $a \times a$ , and $b^{2}$ means $b \times b$ . The lesson on algebraic notation explains these in full; for terms and coefficients the rule is the same.

Example 7: The terms of 21a² − 6ab + b² − 19

For the expression $21a^{2} - 6ab + b^{2} - 19$ , write down
(a) its terms,
(b) the coefficient of $a^{2}$ ,
(c) the coefficient of $ab$ ,
(d) the coefficient of $b^{2}$ ,
(e) its constant term.

Solution

(a) Split at each $+$ and $-$ sign, keeping each sign with its term: $21a^{2}$ , $-6ab$ , $+b^{2}$ , $-19$ .

(b) $21a^{2}$ is $21 \times a^{2}$ , so the coefficient of $a^{2}$ is 21.

(d) $b^{2}$ is $1 \times b^{2}$ , so the coefficient of $b^{2}$ is 1.

(e) The term with no letter is $-19$ , with its minus sign.

(a) The terms are $21a^{2}$ , $-6ab$ , $b^{2}$ and $-19$ . (b) 21 (c) $-6$ (d) 1 (e) The constant term is $-19$ .

Common Mistakes

Reading a letter as a thing. In "Ali buys $a$ apples", $a$ is the number of apples, not "apple". In $60a$ cents, $a$ is the number of apples, and $60a$ is an amount of money: a number of cents.
Writing $4 \times k$ as $k4$ , or as $4 + k$ . Four times $k$ is $4k$ , with the number first; $4 + k$ is 4 more than $k$ .
Subtracting the wrong way round. "Subtract 3 from $n$ " is $n - 3$ , not $3 - n$ .
Dropping the sign of a term. In $2x - 7$ the constant term is $-7$ , not 7; in $5 - 3z$ the coefficient of $z$ is $-3$ , not 3.
Saying that $x$ has no coefficient, or coefficient 0. The coefficient of $x$ is 1, and of $-x$ is $-1$ .
Giving the same letter two values in one expression. In $m + m$ both $m$ s are the same number.

Practice

Try each question before you open its solution. The questions get harder as you go.

Question 1

Write an algebraic expression for each of these.
(a) 9 more than $p$
(b) Subtract 7 from $q$
(c) 6 times $r$
(d) $s$ divided by 5

ShowHide solution

(a) Add 9 to $p$ : $p + 9$ .

(b) Start with $q$ and take 7 away: $q - 7$ .

(d) $s \div 5$ is written $\frac{s}{5}$ .

(a) $p + 9$ (b) $q - 7$ (c) $6r$ (d) $\frac{s}{5}$

Question 2

Write each expression in words.
(a) $h + 12$
(b) $3t$
(c) $w - 1$
(d) $\frac{d}{4}$

ShowHide solution

(a) 12 is added to $h$ : "12 more than $h$ ", or "the sum of $h$ and 12".

(b) $3t$ is $3 \times t$ : "3 times $t$ ".

(d) $\frac{d}{4}$ is $d \div 4$ : " $d$ divided by 4", or " $d$ shared equally among 4".

(a) 12 more than $h$ (b) 3 times $t$ (c) $w$ minus 1 (d) $d$ divided by 4

Question 3

A tray holds 12 eggs. How many eggs are there in
(a) 3 trays,
(b) 10 trays,
(c) $b$ trays?

ShowHide solution

(a) $12 \times 3 = 36$ .

(b) $12 \times 10 = 120$ .

(a) 36 eggs (b) 120 eggs (c) There are $12b$ eggs in $b$ trays.

Question 4

Mrs Goh shares 60 sweets equally among the children at a party. How many sweets does each child get if there are
(a) 5 children,
(b) 12 children,
(c) $c$ children?

ShowHide solution

(a) Sharing equally means dividing: $60 \div 5 = 12$ .

(b) $60 \div 12 = 5$ .

The sweets can be shared out exactly only when $c$ divides 60 exactly (for example 4, 10 or 15 children, but not 7).

(a) 12 sweets (b) 5 sweets (c) Each child gets $\frac{60}{c}$ sweets.

Question 5

At the school bookshop a file costs $3 and a pen costs $2. Siti buys $f$ files and $p$ pens, and pays $(3f + 2p)$ dollars. Ravi says, " $f$ means files and $p$ means pens."
(a) Explain what is wrong with what Ravi says.
(b) What do $3f$ and $2p$ stand for?

ShowHide solution

(a) A letter stands for a number, not for a thing. $f$ is the number of files Siti buys, such as 2 or 5, and $p$ is the number of pens.

(b) Each file costs $3, so $f$ files cost $3 \times f = 3f$ dollars. Each pen costs $2, so $p$ pens cost $2 \times p = 2p$ dollars.

(a) $f$ is the number of files and $p$ is the number of pens, not the files and pens themselves. (b) $3f$ is the cost of the files in dollars, and $2p$ is the cost of the pens in dollars.

Question 6

Say whether each statement is true or false, and give a reason.
(a) In $x + x + 1$ , the two $x$ s can be 2 and 5.
(b) In $x + y$ , $x$ and $y$ can both be 4.
(c) In $p + q$ , $p$ and $q$ must be different numbers.

ShowHide solution

(a) In one expression the same letter stands for the same number, so both $x$ s are 2 or both are 5. False.

(b) Different letters may stand for the same number. True.

(a) False (b) True (c) False

Question 7

A bus travels at $u$ km/h.
(a) The bus travels for 2 hours. What does $2u$ stand for?
(b) A car travels 15 km/h faster than the bus. Write the speed of the car in terms of $u$ .
(c) A cyclist travels at half the speed of the bus. Write the speed of the cyclist in terms of $u$ .

ShowHide solution

(a) $u$ is the speed in km/h and the time is 2 hours. Distance = speed × time, and $2u = u \times 2$ , so $2u$ is the distance travelled in those 2 hours, in kilometres.

(b) 15 km/h faster means add 15 to the speed: $u + 15$ . An expression of more than one term goes in brackets before its unit: $(u + 15)$ km/h.

(a) The distance, in kilometres, that the bus travels in 2 hours. (b) $(u + 15)$ km/h (c) $\frac{u}{2}$ km/h

Question 8

For the expression $4p - q + 7$ , write down its terms, the coefficients of $p$ and $q$ , and its constant term.

ShowHide solution

Split at each sign, keeping the sign with its term: $4p$ , $-q$ , $+7$ .

The coefficient of $p$ is 4. The term $-q$ is $-1 \times q$ , so the coefficient of $q$ is $-1$ .

The term without a letter is 7.

Terms: $4p$ , $-q$ and $7$ . Coefficient of $p$ : 4; coefficient of $q$ : $-1$ . Constant term: 7.

Question 9

For the expression $8 - 5m + n$ , write down its terms, the coefficients of $m$ and $n$ , and its constant term.

ShowHide solution

The first term, 8, has no sign in front, so it is positive. The terms are $8$ , $-5m$ and $+n$ .

The coefficient of $m$ is $-5$ , with its sign. $n = 1 \times n$ , so the coefficient of $n$ is 1.

The constant term is 8: it need not come last.

Terms: $8$ , $-5m$ and $n$ . Coefficient of $m$ : $-5$ ; coefficient of $n$ : 1. Constant term: 8.

Question 10

For the expression $3x^{2} - x + 2xy - 11$ (where $x^{2}$ means $x \times x$ and $xy$ means $x \times y$ ), write down
(a) the number of terms,
(b) the coefficient of $x^{2}$ ,
(c) the coefficient of $x$ ,
(d) the coefficient of $xy$ ,
(e) the constant term.

ShowHide solution

(a) The terms are $3x^{2}$ , $-x$ , $+2xy$ and $-11$ : four terms.

(b) $3x^{2} = 3 \times x^{2}$ , so the coefficient of $x^{2}$ is 3.

(d) $2xy = 2 \times xy$ , so the coefficient of $xy$ is 2.

(e) The term without a letter is $-11$ .

(a) 4 (b) 3 (c) $-1$ (d) 2 (e) $-11$

Question 11

Write an expression in terms of $a$ and $b$ that has three terms, in which the coefficient of $a$ is $-2$ , the coefficient of $b$ is 1 and the constant term is $-6$ .

ShowHide solution

The term in $a$ is $-2 \times a = -2a$ . The term in $b$ is $1 \times b$ , written $b$ . The constant term is $-6$ .

Joining them, each with its sign: $-2a + b - 6$ .

The terms may come in any order, as long as each keeps its sign: $b - 2a - 6$ is the same expression.

$-2a + b - 6$ (or the same three terms in another order, such as $b - 2a - 6$ )

Question 12

Canned drinks are sold in packs of 6. For a CCA outing, Ali buys $k$ packs and gives 2 cans to his teacher.
(a) How many cans does Ali have left if he bought 3 packs?
(b) Write, in terms of $k$ , the number of cans Ali has left.

ShowHide solution

(a) 3 packs hold $6 \times 3 = 18$ cans. After giving away 2, he has $18 - 2 = 16$ cans.

(b) Do the same with $k$ packs: they hold $6 \times k = 6k$ cans, and after giving away 2 he has $(6k - 2)$ cans.

Check with (a): when there are 3 packs, $6k$ is 18 and $6k - 2$ is 16, as before.

(a) 16 cans (b) Ali has $(6k - 2)$ cans left.

Key Points

A letter in algebra stands for a number: one we do not know yet, or any number. Such a letter is a variable.
A letter stands for a number of things or an amount, never for the things themselves: $a$ is the number of apples, not "apple".
An algebraic expression is made of numbers and variables joined by $+$ , $-$ , $\times$ , $\div$ and brackets. It has no equals sign.
To write an expression from words, work a case with numbers first, then do the same to the letter.
$4 \times k$ is written $4k$ , and $k \div 6$ is written $\frac{k}{6}$ .
In one expression the same letter stands for the same number; different letters may stand for different numbers or for the same number.
The terms of an expression are its parts joined by $+$ and $-$ , each with the sign in front of it.
The coefficient of a letter is the number multiplying it, with its sign: $x$ has coefficient 1 and $-x$ has coefficient $-1$ .
The constant term is the term with no letter, with its sign.