Syllabus Edition

First teaching 2023

First exams 2025

|

Equations & Problem Solving (CIE IGCSE Maths: Core)

Revision Note

Test yourself

Forming & Solving Equations

How do I form expressions from words?

  • You can turn common phrases into expressions
    • Use x to represent an unknown value
       
      2 less than "something" x minus 2
      Double "something" 2 x
      5 lots of "something" 5 x
      3 more than "something" x plus 3
      Half of "something" 1 half x space or space x over 2
  • Common words indicating basic operations are:
    • Addition: sum, total, more than, increase
    • Subtraction: difference, less than, decrease
    • Multiplication: product, lots of, times as many, double, triple
    • Division: shared, split, grouped, halved, quartered
  • Brackets help keep the order correct
    • "something" add 1, then multiplied by 3
      • left parenthesis x plus 1 right parenthesis cross times 3 which simplifies to 3 left parenthesis x plus 1 right parenthesis
    • Compare this to "something" multiplied by 3, then add 1
      • x cross times 3 plus 1 which simplifies to 3 x plus 1
  • You may have to choose which unknown to call x
    • If Adam is 10 years younger than Barry, then Barry is 10 years older than Adam
      • Either represent Adam's age as x minus 10 and Barry's age as x
      • Or represent Adam's age as x and Barry's age as x plus 10
    • If Adam's age is half of Barry's age, then Barry's age is double Adam's age
      • So if Adam's age is x then Barry's age is 2 x
      • This makes the algebra easier (rather than using x for Barry's age and 1 half x for Adam's age)

How do I form equations?

  • An equation is a statement with an equals sign that can be solved
  • Try to put in the phrase "is equal to" to see where the equals goes
    • Lisa's age is double Aisha's age and the sum of their ages is ("is equal to") 27 
      • Represent Aisha's age as x and Lisa's age is 2 x
      • The equation is 2 x plus x equals 27
    • When solving, always give the answer in context
      • 3 x equals 27 so x equals 9
      • In context: "Lisa is 18 years old and Aisha is 9 years old"
  • Sometimes you might have two unknown values (x  and y)
    • Use the information to form two simultaneous equations

Worked example

A flowerbed has flowers of three different colours: red, yellow and purple.
The number of yellow flowers is three times the number of red flowers.
The number of purple flowers is 5 more than the number of yellow flowers.
 
If the difference between the number of purple flowers and red flowers is 29, find the number of yellow flowers.

Let the number of red flowers be x
 

x red flowers
 

Multiply this by 3 to get the number of yellow flowers
 

3 x yellow flowers

 

Add 5 to the previous result to get the number of purple flowers
 

3 x plus 5 purple flowers
 

Find the difference between the number of purple and red flowers (purple subtract red, as purple is larger)
 

3 x plus 5 minus x
 

Set the difference equal to 29
 

3 x plus 5 minus x equals 29
 

Simplify the left-hand side (3x - x = 2x)
 

2 x plus 5 equals 29
 

Solve the equation (subtract 5 then divide by 2)
 

table row cell 2 x end cell equals cell 29 minus 5 end cell row cell 2 x end cell equals 24 row x equals cell 24 over 2 end cell row x equals 12 end table
 

This is not the answer to the question asked
The number of yellow flowers is 3x so multiply this answer by 3

There are 36 yellow flowers

Forming Equations from Shapes

How do I form equations from shapes?

  • You need to use all the information given on the diagram and any specific properties of that shape
  • Common 2D shapes that you should know properties for are
    • Triangles: equilateral, isosceles, scalene, right-angled
    • Quadrilaterals: square, rectangle, kite, rhombusparallelogram, trapezium
  • You may be asked about perimeter, area or angles
  • You may be asked about polygons
    • Regular vs irregular polygons
    • Interior vs exterior angles
      • The sum of interior angles is 180(n-2) for an n-sided polygon
  • You may be asked about angles in parallel lines
    • Alternative, corresponding and co-interior
  • You may be asked about 3D shapes involving surface area and volume
    • Prisms have constant cross sections 
      • Volume is cross-section area multiplied by length

Is there anything else that can help?

  • Sketch a diagram if none is given
  • Split up uncommon shapes into the sum or difference of common shapes
  • Look out for important extra information
    • For example, a trapezium "with a line of symmetry"
  • With irregular shapes, assume all angles and lengths are different (unless told otherwise)
  • Put brackets around algebraic expressions when substituting them into geometric properties

Forming and solving an equation from an irregular polygon

Examiner Tip

  • Read the question carefully - does it want an angle? perimeter? total area? curved surface area? etc.
  • For surface area and volume questions, check the list of formulas given in the exam.

Worked example

A rectangle has a length of 3 x plus 1 cm and a width of 2 x minus 5 cm.

Its perimeter is equal to 22 cm.

(a)
Use the above information to find the value of x.

The perimeter of a rectangle is 2 × length + 2 × width

2 left parenthesis 3 x plus 1 right parenthesis plus 2 left parenthesis 2 x – 5 right parenthesis

Expand the brackets

space 6 x plus 2 plus 4 x minus 10

Simplify by collecting like terms

10 x minus 8

This perimeter is 22, so set this expression equal to 22

10 x – 8 equals 22

Solve this equation by adding 8 then dividing by 10

table row cell 10 x end cell equals cell 22 plus 8 end cell row cell 10 x end cell equals 30 row x equals cell 30 over 10 end cell row x equals 3 end table

bold italic x bold equals bold 3

(b)
Find the area of the rectangle.

The area of a rectangle is its length multiplied by is width
Substitute the value of x  from part (a) into the length and width given in the question

length is 3 × 3 + 1 = 10

width is 2 × 3 - 5 = 1

Find the area (multiply length by width)

10 × 1

Include the correct units for area

Area = 10 cm2

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.