Constructing Equations

Turning words into expressions and formulas, forming and solving equations, and equations from perimeters and angles.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 108 real past-paper questions below

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From Words to Algebra

These questions are written in words. Your job is to turn the words into algebra, one phrase at a time.

Words	Algebra
5 more than x	$x + 5$
4 fewer than x	$x - 4$
twice as many as x	$2 x$
p pens at $3 each	$3 p$
n bags cost $d, so one bag costs	$\frac{d}{n}$
$48 on books at $x each, so the number of books is	$\frac{48}{x}$
12 km at x km/h takes (hours)	$\frac{12}{x}$
n sweets, give away a fifth, so you have left	$n - \frac{n}{5}$
think of n, square it, then subtract 55	$n^{2} - 55$

Rule

Price × number gives the cost of a group. Add the groups for the total. To find the cost of one, or how many, divide.

Key idea

An expression has no equals sign. A formula starts with a letter and an equals sign, like $T = \dots$ .

Key idea

A bat and a ball cost $1.10 together. The bat costs $1 more than the ball. How much is the ball? Most people say 10 cents! Let the ball cost x dollars and write it out.

Worked example

1
$x + (x + 1) = 1.10$ — The ball is x and the bat is $1 more, x + 1. Together they cost 1.10.
2
$2 x + 1 = 1.10$ — Collect the x's: x + x = 2x. This is the line people skip, and why they get it wrong.
3
$2 x = 0.10$ — Take 1 from both sides.
4
$x = 0.05$ — Divide both sides by 2. The ball costs 5 cents and the bat $1.05.

Writing a Formula

“Write a formula for T” is the most common question here, worth up to 3 marks.

Adult tickets cost $8 and child tickets cost $5. A group buys a adult and c child tickets. The total cost is T dollars.

Worked example

1
$8 a$ — the adults: $8 each, a of them
2
$5 c$ — the children: $5 each, c of them
3
$T = 8 a + 5 c$ — add the groups; start with “T =”

Common mistake

Writing only $8 a + 5 c$ . That is an expression. The question asked for a formula, so it needs $T =$ in front.

Forming and Solving an Equation

When the question gives a total, write an equation and solve it.

Ali has x sweets. Ben has 3 times as many as Ali. Cara has twice as many as Ali. They have 60 sweets in total. How many does Ben have?

Worked example

1
$x + 3 x + 2 x = 60$ — Ali, Ben and Cara add to the total, 60.
2
$6 x = 60$ — Collect like terms: x + 3x + 2x = 6x.
3
$x = 10$ — Divide both sides by 6. Ali has 10.
4
$3 x = 30$ — Ben has 3 × 10 = 30. Answer the question that was asked.

Common mistake

Stopping at x = 10. The question asked about Ben, so finish with 3 × 10 = 30.

Key idea

Write the equation down before you solve it. The first mark is usually for the equation itself, so you keep it even if you slip later. An answer with no working can lose most of the marks.

Two Expressions That Are Equal

Sometimes there is no total. Instead the question says two things are “equal” or “the same”. Then one expression = the other.

A rectangle has sides (x + 3) cm and (x − 1) cm. An equilateral triangle has sides (x + 5) cm. Their perimeters are equal. Find x.

Worked example

1
$2 (x + 3) + 2 (x - 1)$ — The rectangle's perimeter: two of each side.
2
$= 2 x + 6 + 2 x - 2$ — Expand each bracket. 2 × (−1) = −2, so keep the minus.
3
$= 4 x + 4$ — Collect: 2x + 2x = 4x and 6 − 2 = 4.
4
$3 (x + 5) = 3 x + 15$ — The triangle's perimeter: three equal sides, then expand.
5
$4 x + 4 = 3 x + 15$ — The perimeters are equal, so the two expressions are equal.
6
$x + 4 = 15$ — Take 3x from both sides, so x is on one side only.
7
$x = 11$ — Take 4 from both sides.
8
$4 (11) + 4 = 48, 3 (11) + 15 = 48$ — Check: both perimeters are 48 cm, so x = 11 is right.

Rule

“Is equal to”, “the same as”, “equal perimeters”: write expression = expression. Then collect the x's on one side.

Shapes: Perimeter and Area

Harder questions put the algebra on a shape. Write the perimeter or the area in x, set it equal to what you are told, then solve.

A rectangle has sides (2x + 3) cm and (x − 1) cm. Its perimeter is 28 cm. Find its area.

Worked example

1
$2 (2 x + 3) + 2 (x - 1) = 28$ — A rectangle has two long sides and two short sides. They add to 28.
2
$4 x + 6 + 2 x - 2 = 28$ — Expand each bracket. 2 × (−1) = −2, so keep the minus.
3
$6 x + 4 = 28$ — Collect: 4x + 2x = 6x and 6 − 2 = 4.
4
$6 x = 24$ — Take 4 from both sides.
5
$x = 4$ — Divide both sides by 6.
6
$2 (4) + 3 = 11, 4 - 1 = 3$ — Put x = 4 back into each side. The sides are 11 cm and 3 cm.
7
$11 \times 3 = 33 cm^{2}$ — The question asked for the area, not x. Area = length × width.

Sometimes the area itself is the equation. A rectangle is 4 cm wide and (x + 3) cm long. Its area is 36 cm². Find its length.

Worked example

1
$4 (x + 3) = 36$ — Area = width × length, and the area is 36.
2
$4 x + 12 = 36$ — Expand the bracket: 4 × x = 4x and 4 × 3 = 12.
3
$4 x = 24$ — Take 12 from both sides.
4
$x = 6$ — Divide both sides by 4.
5
$6 + 3 = 9 cm$ — The question asked for the length, so put x = 6 into x + 3. Check: 4 × 9 = 36.

Key idea

Perimeter: add all the sides. Area: multiply length × width. Pick the one the question gives you.

Angle Equations

Angle questions give each angle in x. You need the right angle fact to know what they add to.

Rule

Angles in a triangle add to 180°. On a straight line: 180°. Around a point: 360°. In a quadrilateral: 360°. In a parallelogram, two next-door angles add to 180°.

A parallelogram has two next-door angles of (2x + 15)° and (x + 30)°. Find both angles.

Worked example

1
$(2 x + 15) + (x + 30) = 180$ — Next-door angles in a parallelogram add to 180°. Say this fact in your answer.
2
$3 x + 45 = 180$ — Collect: 2x + x = 3x and 15 + 30 = 45.
3
$3 x = 135$ — Take 45 from both sides.
4
$x = 45$ — Divide both sides by 3.
5
$2 (45) + 15 = 105, 45 + 30 = 75$ — Put x back in. The angles are 105° and 75°. Check: 105 + 75 = 180.

Common mistake

Giving x = 45 as the answer. The question asked for the angles, so work out 105° and 75°.

A Fixed Charge, Then a Rate

Hire and taxi questions charge one price to start, then a price for each extra hour or km. Watch the word “extra”.

A bike costs $12 for the first hour, then $4.50 for each extra hour. Mo paid $39. For how many hours, h, did he hire the bike?

Worked example

1
$12 + 4.5 (h - 1) = 39$ — The first hour is $12. The first hour is already paid for, so only h − 1 hours are extra.
2
$4.5 (h - 1) = 27$ — Take 12 from both sides.
3
$h - 1 = 6$ — Divide both sides by 4.5.
4
$h = 7$ — Add 1 to both sides. Mo hired the bike for 7 hours.
5
$12 + 4.5 \times 6 = 39$ — Check: the first hour plus 6 extra hours costs $39.

Common mistake

Writing 12 + 4.5h. That charges the first hour twice and gives h = 6. Only the hours after the first are extra, so use (h − 1).

Mixed Units: Convert First

Some questions mix units, like prices in cents and a note in dollars. Change everything to one unit before you write the equation.

Pens cost 40 cents each. Lea buys p pens and pays with a $20 note. Her change is $4.40. Find p.

Worked example

1
$$20 = 2000 cents$ — The pens are in cents, so change the $20 to cents: 20 × 100.
2
$40 p$ — The pens cost 40p cents: 40 cents each, p of them.
3
$2000 - 40 p$ — Change = what she paid − what the pens cost.
4
$2000 - 40 p = 440$ — The change was $4.40, which is 440 cents.
5
$2000 = 440 + 40 p$ — Add 40p to both sides, so p is positive.
6
$1560 = 40 p$ — Take 440 from both sides.
7
$p = 39$ — Divide both sides by 40. Lea bought 39 pens.

Rule

$1 = 100 cents. 1 m = 100 cm. 1 litre = 1000 ml. Convert first, then write the equation.

Common mistake

Writing 20 − 40p. That mixes dollars and cents: it gives p = 0.39, and you cannot buy 0.39 of a pen.

Which Method? Read the Question

The last line of the question tells you what to write.

You see	You do
“Write an expression”	Algebra with no equals sign
“Write a formula for T”	Start with T =, then add the groups
A total is given	Write an equation and solve it
“Equal” or “the same as”	Expression = expression
A shape	Add sides or angles, or multiply for area, then set equal
First hour, then each extra hour	Fixed price + rate × (h − 1)
Cents and dollars, or cm and m	Change to one unit first

Key idea

After solving, read the question again. It often wants something worked out from x, not x itself.

Key idea

If the equation you form has an x² in it, it becomes a quadratic. The lesson Equations That Become Quadratics shows how to finish it.

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