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Checkpoint 8 Maths · Algebra · Expressions and formulae

Writing Expressions and Formulae, and Changing the Subject

Translate between words and algebraic expressions or formulae (including fractional coefficients), and use inverse operations to change the subject of a formula.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum
  • Change a situation in words into an algebraic expression, and an expression back into words — including expressions with fractions, such as ½n − 3 or 2n/5 (8Ae.04)
  • Change a situation in words into a formula with mixed operations, and use inverse operations to change the subject of a formula, e.g. make x the subject of y = 3x − 5 (8Ae.05)

Algebra is a language. On this page you learn to translate between English and algebra — turning sentences into expressions and formulae, and reading algebra back as words. Then you learn to rearrange a formula so that a different letter is on its own: this is called changing the subject, and it uses the same inverse operations you use to solve equations.

Key words

Word / phraseMeaning / use
expressionمقدار جبري — terms with no equals sign, e.g. 3n − 8
formulaقانون / صيغة — a rule that links quantities, e.g. C = 5 + 3h
subject of a formulaموضوع القانون — the letter on its own on one side; in C = 5 + 3h the subject is C
change the subject (rearrange)تغيير موضوع القانون — rewrite the formula so another letter is on its own
inverse operationالعملية العكسية — the operation that undoes another: + and −, × and ÷, squaring and square root
coefficientمعامل — the number multiplying a letter; in ¾n the coefficient is ¾
substituteيعوّض — replace letters by numbers
sum, difference, productمجموع، فرق، حاصل ضرب — the answers to +, −, ×

1. From words to expressions

WordsExpressionWatch out
5 more than nn + 5
5 less than nn − 5not 5 − n
n less than 5 / subtract n from 55 − nthe order is reversed in English
twice n / double n2n
the product of x and yxy
a third of nn/3 or ⅓nboth are correct
three-quarters of n3n/4 or ¾nboth are correct
multiply n by 4, then add 14n + 1
add 1 to n, then multiply by 44(n + 1)brackets show 'do this first'
half of the sum of a and b(a + b)/2not a + b/2
the square of n, then multiply by 33n²not (3n)²
n divided by 2, then subtract 3n/2 − 3 or ½n − 3
subtract 3 from n, then divide by 2(n − 3)/2

Key idea: the order of the instructions decides where the brackets go. If an addition or subtraction must happen before a multiplication or division, it needs brackets (or a fraction line over it).

In context. A pen costs p cents and a ruler costs 30 cents more: ruler = p + 30. Ali has x sweets and gives away a quarter of them: he has ¾x left. Sara earns $d per day and saves two-fifths of it: she saves 2d/5 (or ⅖d) per day. In 7 days she saves 7 × 2d/5 = 14d/5.

2. From expressions to words

Read the expression in the order of operations (brackets, indices, ×/÷, +/−) and say each step:

ExpressionIn words
3x − 7multiply x by 3, then subtract 7
3(x − 7)subtract 7 from x, then multiply by 3
x/4 + 2divide x by 4, then add 2
(x + 2)/4add 2 to x, then divide by 4
⅔xtwo-thirds of x (multiply by 2, divide by 3)
5x²square x, then multiply by 5

3. Writing formulae

A formula has a subject and an equals sign. Steps: (1) choose letters and say what each one means (with units); (2) write a worked example with numbers; (3) replace the numbers by letters.

  • Hiring a bike costs $5 plus $3 for each hour. For 4 hours: 5 + 3 × 4 = $17. Formula: C = 5 + 3h, where C is the cost in dollars and h the number of hours.
  • A taxi charges $2.50 plus $0.80 per kilometre: C = 2.5 + 0.8k.
  • Average speed s is distance d divided by time t: s = d/t.
  • The number of fence posts P for a straight fence of length L metres, with a post every 2 m and one extra at the start: P = L/2 + 1.
  • The perimeter of a rectangle of length l and width w: P = 2l + 2w or P = 2(l + w).
(3x + 2) cmx cmNOT TO SCALE
A rectangle with length (3x + 2) cm and width x cm, used in Worked example 4.

4. Changing the subject of a formula

To make a different letter the subject, undo what has been done to it, in reverse order, using inverse operations — and do the same thing to both sides. A function machine helps you see the order.

Forwards: start with x, end with yx× 3− 5yBackwards (inverse): start with y, end with xx÷ 3+ 5yy = 3x − 5 → x = (y + 5) ÷ 3
In y = 3x − 5, x is multiplied by 3 and then 5 is subtracted. To get x back, add 5 and then divide by 3, so x = (y + 5)/3.
  1. Find the letter you want as the new subject.
  2. List the operations done to it, in order (use BIDMAS).
  3. Undo them one by one in reverse order, doing the same to both sides.
  4. Write the new subject on the left: x = …
  5. Check by substituting numbers into both versions.
FormulaMake … the subjectSteps (both sides)Answer
y = x + 7x− 7x = y − 7
P = 4ss÷ 4s = P/4
y = 3x − 5x+ 5, then ÷ 3x = (y + 5)/3
y = x/4 + 2x− 2, then × 4x = 4(y − 2)
A = ½bhh× 2, then ÷ bh = 2A/b
P = 2l + 2wl− 2w, then ÷ 2l = (P − 2w)/2
v = u + ata− u, then ÷ ta = (v − u)/t
y = 10 − xx+ x, then − yx = 10 − y

Check example: y = 3x − 5. If x = 4, then y = 7. Put y = 7 into x = (y + 5)/3: (7 + 5)/3 = 4 ✓. When the new formula gives back the number you started with, your rearranging is correct.

تغيير موضوع القانون

اسأل نفسك: إيه اللي حصل للحرف اللي عايزه؟ وبالترتيب؟ وبعدين اعكس كل خطوة بالعكس: الجمع يتشال بالطرح، والضرب يتشال بالقسمة — وابدأ بآخر عملية حصلت. وأي حاجة تعملها في ناحية، اعملها في الناحية التانية.

Worked examples

Worked example 1 — Write an expression for each. (a) Multiply n by 3, then subtract 8. (b) Subtract 8 from n, then multiply by 3. (c) Two-thirds of n, add 1. [3]

  • (a) 3n − 8
  • (b) The subtraction happens first, so it needs brackets: 3(n − 8)
  • (c) ⅔n + 1 (also accept 2n/3 + 1)

Common mistake: writing 3n − 8 for (b). Without brackets, the multiplication would be done first.

Worked example 2 — Amira is x years old. Her brother is 4 years younger than Amira. Her mother is three times as old as Amira. (a) Write an expression for her brother's age. (b) Write an expression for her mother's age. (c) Write and simplify an expression for the total of their three ages. [3]

  • (a) x − 4
  • (b) 3x
  • (c) x + (x − 4) + 3x = 5x − 4

Worked example 3 — A taxi company charges a fixed $3 plus $1.20 for each kilometre. (a) Write a formula for the cost C dollars of a journey of k kilometres. (b) Find the cost of a 15 km journey. (c) A journey costs $12.60. How far was it? [4]

  • (a) C = 3 + 1.2k (or C = 1.2k + 3).
  • (b) C = 3 + 1.2 × 15 = 3 + 18 = $21.
  • (c) Undo the operations: subtract 3, then divide by 1.2. k = (12.60 − 3) ÷ 1.2 = 9.6 ÷ 1.2 = 8 km. (Or rearrange first: k = (C − 3)/1.2.)

Worked example 4 — The rectangle in the diagram above has length (3x + 2) cm and width x cm. (a) Write a formula for its perimeter P. Simplify your answer. (b) Make x the subject of your formula. (c) Find x when P = 60. [4]

  • (a) P = 2(3x + 2) + 2x = 6x + 4 + 2x → P = 8x + 4.
  • (b) Operations on x: × 8, then + 4. Undo: − 4, then ÷ 8 → x = (P − 4)/8.
  • (c) x = (60 − 4)/8 = 56/8 = 7. Check: length 23, width 7 → 2 × 23 + 2 × 7 = 60 ✓

Worked example 5 — Make x the subject of y = x/5 + 2. [2]

  • Operations on x: ÷ 5, then + 2.
  • Subtract 2 from both sides: y − 2 = x/5.
  • Multiply both sides by 5: x = 5(y − 2) (or x = 5y − 10).
  • Check: x = 15 gives y = 3 + 2 = 5, and 5(5 − 2) = 15 ✓

Worked example 6 — (a) Make h the subject of A = ½bh. (b) Make l the subject of P = 2l + 2w. [4]

  • (a) Multiply both sides by 2: 2A = bh. Divide both sides by b: h = 2A/b.
  • (b) Subtract 2w: P − 2w = 2l. Divide by 2: l = (P − 2w)/2 (also correct: l = P/2 − w).

Common mistake in (b): writing l = P − 2w ÷ 2. That means only 2w is divided by 2. Use a bracket or a long fraction line.

Worked example 7 (challenge) — The formula C = 5(F − 32)/9 changes a temperature in °F to °C. Make F the subject. [3]

  • Operations on F, in order: − 32, × 5, ÷ 9.
  • Undo ÷ 9: 9C = 5(F − 32).
  • Undo × 5: 9C/5 = F − 32.
  • Undo − 32: F = 9C/5 + 32.
  • Check: C = 100 → F = 900/5 + 32 = 180 + 32 = 212 ✓ (water boils at 100 °C = 212 °F).
Common mistakes
  • Reversing 'less than': '5 less than n' is n − 5, not 5 − n.
  • Forgetting brackets: 'add 1 to n, then multiply by 4' is 4(n + 1), not 4n + 1.
  • Writing a formula without the subject, e.g. '5 + 3h' when asked for a formula for C. Write C = 5 + 3h.
  • Undoing operations in the wrong order: for y = 3x − 5, dividing by 3 first gives x = y/3 + 5, which is wrong.
  • Doing an operation to only one term: from y − 2 = x/5, multiplying gives 5(y − 2) = x, not 5y − 2.
  • Dividing only part of an expression: (P − 2w)/2 is not P − 2w/2.
  • Not checking — substituting a pair of numbers takes 20 seconds and catches most errors.
How Checkpoint asks this
  • 'Write an expression for…' (no equals sign) versus 'Write a formula for…' (must start 'C = …'). Read the command.
  • 'Draw lines to match each description with its expression' — check where the brackets go.
  • Context questions with ages, costs, perimeters and fixed charges, often in several parts that build on each other.
  • 'Make x the subject of the formula' — show each step; a correct first step can earn a method mark.
  • Fractional coefficients: ½n, ¾x and 2n/5 are all acceptable ways of writing the answer.
  • 'Rearrange and then find…' — rearrange first, then substitute; or substitute first and solve. Both get the marks if the working is clear.

Practice

Remember
  • The order of the words decides the brackets: 'add 1, then × 4' = 4(n + 1).
  • '5 less than n' = n − 5; 'subtract n from 5' = 5 − n.
  • A quarter of n = n/4 = ¼n; fractional coefficients are fine.
  • A formula has a subject: C = 5 + 3h. An expression does not.
  • Change the subject: undo the operations in reverse order, doing the same to both sides; check with numbers.
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