GCSE Maths · Substitution and Formulae · Higher · Lesson 8.1

GCSE Maths: Letters, expressions and formulae (Higher)

Part of the GCSE Maths substitution and formulae chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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8.1

Letters, expressions and formulae

Edexcel 1MA1: A3, A21
Grade 1-4

Algebra uses letters to stand for numbers. A letter whose value can change is called a variable. If a bag holds n sweets, then n might be 20 today and 25 tomorrow. Writing a rule once with a letter saves us writing it again for every number.

Four words are used all the time in algebra. Exam questions expect you to know them without being told.

Word	Meaning	Example
Expression	Numbers and letters joined by +, −, ×, ÷. There is no equals sign.	3n + 5
Equation	Two expressions joined by an equals sign. It is true for only some values of the letter.	3n + 5 = 20 (true when n = 5)
Formula	A rule linking real quantities, written with an equals sign. You use it again and again with different numbers.	A = lw (area = length × width)
Identity	Two expressions that are equal for every value of the letter. We write ≡.	2(n + 3) ≡ 2n + 6

A term is one part of an expression, separated from the next by a + or − sign. In 3n + 5 the terms are 3n and 5.

Writing algebra in short form

To keep algebra tidy we do not write the multiplication sign. The rules are:

Algebra shorthand

3 × n is written 3n: the number goes first and there is no × sign.
n × n is written n² (say “n squared”), and n × n × n is n³.
a × b is written ab, with the letters in alphabetical order.
n ÷ 4 is written as a fraction: n4.
2 × (n + 1) is written 2(n + 1).

Turning words into algebra

Imagine Sam has n sweets and Tia has 5 more than Sam. Draw it as a bar. Tia has Sam’s amount n and then 5 extra, so Tia has n + 5 sweets. Drawing a bar like this is a good way to see what the words mean before you write any algebra.

Sam has n sweets. Tia has the same again plus 5 more, so n + 5.

Some words always point to the same operation. Use this table to translate, but always read the whole sentence.

Words in the question	Operation	Example in algebra
more than, plus, sum, total, increased by	add (+)	7 more than n is n + 7
less than, minus, fewer, decreased by, take away	subtract (−)	7 less than n is n − 7
times, multiplied by, product, twice, double, triple	multiply (×)	double n is 2n
divided by, shared equally between, half of	divide (÷)	half of n is n2

Finally, an expression can be pictured as a function machine: you put a number in, the machine does the operations in order, and a number comes out. The expression 3n + 2 is the machine “multiply by 3, then add 2”.

A function machine for 3n + 2. Put in n = 4: 4 × 3 = 12, then 12 + 2 = 14.

Watch out

“Less than” swaps the order. “5 less than n” means start from n and take 5 away: n − 5. It is not 5 − n. Test it: if n = 8, then 5 less than 8 is 3, and 8 − 5 = 3. Also remember 2n (double n) is not the same as n² (n × n): with n = 5 they are 10 and 25.

Worked example 8.1.1

Write an expression for “7 less than n”.

Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.

Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.

“7 less than” “Less than” means take away. The 7 is taken away from the number.
− 7
“n” The letter n stands for the number we start with. It comes first because we start from it.
n
Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.
n − 7

Check. Test with a number. Let n = 10. In words: 7 less than 10 = 3. The expression gives 10 − 7 = 3. They agree. ✓

Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.

Worked example 8.1.2

A pen costs p pence. A notebook costs 90 pence. Write an expression for the total cost, in pence, of 4 pens and one notebook.

Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.

Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.

“A pen costs p pence” We let the letter p stand for the price of one pen, in pence.
p
“4 pens” 4 pens cost 4 lots of p, which is 4 × p. In algebra shorthand that is 4p.
4p
“and a notebook costing 90 pence” “And” tells us to add the notebook. The notebook is a fixed 90 pence.
+ 90
Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.
4p + 90

Check. Test with a number. Let p = 30. In words: 4 pens at 30p each is 120p, plus 90p = 210. The expression gives 4 × 30 + 90 = 120 + 90 = 210. They agree. ✓

Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.

Worked example 8.1.3

Write down whether each is an expression, an equation, a formula or an identity: (a) 5n − 2 (b) 5n − 2 = 18 (c) 2(n + 3) ≡ 2n + 6.

Method. For each one, look for an equals sign, then decide which word fits.

(a) 5n − 2 There is no equals sign.
It is an expression. It is a phrase made of the terms 5n and −2.
(b) 5n − 2 = 18 There is an equals sign, but it is only true for one value of n. Try n = 1: 5 × 1 − 2 = 3, not 18. Try n = 4: 5 × 4 − 2 = 18, which works.
It is an equation. Solving it gives n = 4.
(c) 2(n + 3) ≡ 2n + 6 It uses ≡. Expanding the bracket: 2 × n = 2n and 2 × 3 = 6, so the left side is exactly 2n + 6 for every n.
It is an identity.

Check. For (c) try n = 10: left = 2(10 + 3) = 26 and right = 2 × 10 + 6 = 26. They agree. ✓

Watch out. An equals sign alone does not make something an equation. Check whether it is true for all values (identity) or only some (equation).

Worked example 8.1.4

n is a whole number. Write an expression for the sum of three consecutive whole numbers, starting with n, and simplify it. Explain why the answer is always a multiple of 3.

Method. Consecutive numbers go up by 1 each time. Add them, collect like terms, then look for a factor of 3.

Write the three numbers. Consecutive whole numbers follow each other: each is 1 more than the one before.
n, n + 1, n + 2
Add them. The sum means add all three.
n + (n + 1) + (n + 2)
Collect like terms. Add the n terms together (n + n + n = 3n) and add the numbers together (1 + 2 = 3).
3n + 3
Factorise. Both terms have a common factor of 3.
3n + 3 = 3(n + 1)
Explain. Anything written as 3 × (a whole number) is a multiple of 3, and n + 1 is a whole number.
So the sum is always a multiple of 3.

Check. Try n = 4: 4 + 5 + 6 = 15 = 3 × 5. Try n = 10: 10 + 11 + 12 = 33 = 3 × 11. ✓

Watch out. Do not stop at 3n + 3. To explain “multiple of 3” you must show the factor of 3, by writing 3(n + 1).

Exercise 8A Words, expressions and formulae

Non-calculator

FWrite down whether each of these is an expression, an equation, a formula or an identity.

(a)4n − 7

(b)4n − 7 = 21

(d)2(l + w)

(e)P = 2l + 2w

(f)5 + 2x

Done

FWrite each of these in the shortest algebra form.

(a)3 × n

(b)n × n

(d)b × a × 3

(e)x ÷ 5

(f)2 × (n + 4)

Done

FWrite an expression for each statement. Use the letter n for the number.

(a)5 more than n

(b)6 times n

(d)n shared equally between 4 people

(e)double n, then add 3

(f)take n away from 12

Done

FPens cost 35 pence each and a ruler costs 60 pence. Write an expression, in pence, for the cost of:

(a)n pens

(b)n pens and one ruler

(c)the change from £5 when buying n pens and one ruler

Done

FAli is a years old. Ben is 3 years older than Ali. Cara is half as old as Ali. Write an expression for:

(a)Ben’s age

(b)Cara’s age

(c)the total of their three ages (you do not need to simplify)

Done

RDan says, “4 more than n is 4n.” Show, using n = 5, that Dan is wrong, and write the correct expression.

Done

PSA taxi charges £3 to start, plus £2 for every mile. The cost is £C for a journey of m miles.

(a)Write a formula for C in terms of m.

(b)Use your formula to work out the cost of a 5 mile journey.

Done

FWrite an expression for each description.

(a)Add 3 to n, then double the result

(b)Double n, then add 3

(c)Multiply n by itself, then add 3

(d)Add 3 to n, then multiply the result by itself

Done

FThe diagram shows a triangle. The lengths are in centimetres. Write down and simplify an expression for the perimeter.

Done

PSAmir has £a. Bea has £5 more than Amir. Cal has twice as much as Amir. Write an expression for the amount that:

(a)Bea has

(b)Cal has

(c)all three have altogether, in its simplest form

Done

PSn is a whole number. Write an expression for:

(a)the next whole number after n

(b)the whole number just before n

(c)the next even number after the even number 2n

Done

RTia says that 3(n + 2) is the same as 3n + 2. Test her idea with n = 1 and n = 4. Is she right?

Done

PSA cinema sells adult tickets for £a each and child tickets for £c each. A family buys 3 adult tickets and 2 child tickets. Write an expression for the total cost in pounds.

Done

Rn is a whole number. Explain why 2n is always even and why 2n + 1 is always odd.

Done

Rn is an odd number. Show that the sum of n and the next two odd numbers is always a multiple of 3.

Done

RA rectangle has width w cm. Its length is 3 cm more than twice its width. Write and simplify an expression for the perimeter of the rectangle. Explain why the perimeter is always a multiple of 6.

Done