← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 3 · pp. 24–29

Calculations and order

Two kinds of order: putting numbers in order of size, with the symbols that compare them, and doing the parts of a calculation in the right order. Get the second one wrong and 2 + 3 × 4 comes out as 20 instead of 14.

Learning objectives

By the end of this chapter you can:

  1. Use the symbols =, ≠, <, ≤, > and ≥, and write a sentence as an inequality.
  2. Show an inequality on a number line with open and filled circles, and read one back.
  3. Put decimals, fractions and measurements in different units in order of size.
  4. Use the order of operations (brackets, indices, ×÷, +−) by hand and on a calculator.

Book exercises: 3.1 to 3.5 and Student assessments 1 to 3 (pp. 24–29).

Next: symbols and inequalities →

Symbols and inequalities

🔑 Key ideas

  • The pointed end of < or > always points at the smaller number.
  • An extra line underneath (≤, ≥) adds “or equal to”.
  • On a number line: filled circle ● = included (≤, ≥); open circle ○ = not included (<, >).
SymbolMeansExampleCircle
=is equal to48 ÷ 6 = 54 ÷ 9—
≠is not equal to7 × 2 ≠ 8 + 7—
>is greater thanx > 3○ open
≥is greater than or equal tox ≥ −7● filled
<is less thanx < 5○ open
≤is less than or equal ton ≤ 11● filled

An inequality can be read from either side: 5 > x says the same as x < 5. Flip the whole statement and the symbol flips with it.

2 ≤ x < 5 1 2 3 4 5 6 x ≥ −7 −8 −7 −6 −5 −4
Top: 2 is included (filled), 5 is not (open). Bottom: −7 is included, and the arrow shows the values carry on for ever.

📘 Example 1 · A football team may have 7 to 11 players on the pitch

  1. Choose a letter for the unknown: let n be the number of players.
  2. “At least 7” means 7 is allowed: n ≥ 7. “At most 11” means 11 is allowed: n ≤ 11.
  3. Join them, smallest on the left: 7 ≤ n ≤ 11.
  4. On a number line: filled circles at 7 and 11, joined by a line. (Players come in whole numbers, so only 7, 8, 9, 10, 11 count.)

✏️ Try it

  1. Put =, > or < between: 5 × 10 … 7² and 80 cm … 1 m.
  2. Write as an inequality: “A farmer has between 350 and 400 apples on each tree.”
  3. Rewrite −7 ≤ x the other way round.
Answers

1. 50 > 49 and 80 cm < 1 m · 2. 350 < a < 400 (or ≤ if 350 and 400 count) · 3. x ≥ −7

Next: ordering numbers →

Putting numbers in order of size

🔑 Key ideas

  • Decimals: pad them to the same number of decimal places, then compare left to right.
  • Fractions: turn each into a decimal (or give them a common denominator).
  • Measurements: change everything into the same unit first.

📘 Example 2 · Order 0.6, 0.66, 0.606, 0.06, smallest first

  1. Pad every number to 3 d.p.: 0.600, 0.660, 0.606, 0.060.
  2. Now compare them like whole numbers: 600, 660, 606, 60.
  3. Order: 60, 600, 606, 660 → 0.06, 0.6, 0.606, 0.66.
0 0.5 1 2/190.11 7/180.39 6/130.46 1/3 4/50.8
Fractions placed on a number line by their decimal values: 2/19 < 1/3 < 7/18 < 6/13 < 1/2 < 4/5.

📘 Example 3 · Order 2 m, 60 cm, 800 mm, 180 cm, 0.75 m, smallest first

  1. Choose one unit, say cm: 2 m = 200 cm, 800 mm = 80 cm, 0.75 m = 75 cm.
  2. The list is now 200, 60, 80, 180, 75 (all cm).
  3. Order: 60, 75, 80, 180, 200 → 60 cm, 0.75 m, 800 mm, 180 cm, 2 m.

💡 Useful facts: 1 m = 100 cm = 1000 mm, 1 kg = 1000 g, 1 l = 100 cl = 1000 ml = 1000 cm³.

✏️ Try it

  1. Order, largest first: 4 kg, 3500 g, ¾ kg, 700 g, 1 kg.
  2. Order, smallest first: 1 l, 430 ml, 800 cm³, 120 cl, 150 cm³.
Answers

1. 4 kg, 3500 g, 1 kg, ¾ kg, 700 g · 2. 150 cm³, 430 ml, 800 cm³, 1 l, 120 cl

Next: order of operations →

The order of operations

🔑 Key ideas

  • Work in this order: Brackets, Indices, Division and Multiplication, Addition and Subtraction (BIDMAS).
  • × and ÷ have the same priority: do them left to right. The same for + and −.
  • A scientific calculator follows BIDMAS. For a fraction line, put brackets round the top and the bottom.
B I D M A S 1 Brackets ( ) 2 Indices x², √ 3 × ÷ left → right 4 + − left → right
Work from the top of the pyramid down. Operations on the same level are done left to right.

📘 Example 4 · 2 + 3 × 4 and (2 + 3) × 4

2 + 3 × 4

  1. No brackets or indices. Multiplication comes before addition: 3 × 4 = 12.
  2. 2 + 12 = 14.

(2 + 3) × 4

  1. Brackets first: 2 + 3 = 5.
  2. 5 × 4 = 20.

📘 Example 5 · 3 + 7 × (2 − 4)

  1. Brackets: 2 − 4 = −2. The sum is now 3 + 7 × (−2).
  2. Multiply: 7 × (−2) = −14. The sum is now 3 + (−14).
  3. Add: 3 − 14 = −11.

📘 Example 6 · on a calculator

  1. The fraction line groups the top and the bottom, so each needs brackets.
  2. Type ( 12 + 9 ) ÷ ( 10 − 3 ) =.
  3. Answer: 21 ÷ 7 = 3. Without brackets the calculator would do 12 + 0.9 − 3 = 9.9.

📘 Example 7 · Put brackets in to make 9 + 2 × 4 = 44 correct

  1. Without brackets: 9 + 8 = 17. Too small, so the addition must happen first.
  2. Try (9 + 2) × 4 = 11 × 4 = 44. ✔
  3. Answer: (9 + 2) × 4 = 44.

🧪 Try it: step-by-step solver

Type a calculation with numbers, + − × ÷ (or * /), ^ for powers, and brackets.

    ✏️ Try it, in your head first

    1. 12 ÷ 3 + 6 × 5
    2. −14 × 2 − 16 ÷ 2
    3. Put brackets in: 12 ÷ 4 − 2 + 6 = 12
    Answers

    1. 4 + 30 = 34 · 2. −28 − 8 = −36 · 3. 12 ÷ (4 − 2) + 6 = 12

    Next: summary and quiz →

    Chapter summary

    • < and > point at the smaller number; ≤ and ≥ include “equal to”. Flip an inequality and the symbol flips too.
    • On a number line, ● filled means the end value is included and ○ open means it is not.
    • To order decimals, pad them to the same number of places. To order fractions, change them to decimals. To order measurements, use one unit.
    • BIDMAS: brackets, indices, then × and ÷ left to right, then + and − left to right.
    • On a calculator, put brackets round the top and the bottom of any fraction.

    📄 Practice questions

    The book's Exercises 3.1 to 3.5 and Student assessments 1 to 3 are in the printable assignment for this chapter.

    Mini-quiz

    1. What is 2 + 3 × 4?
    2. What is (2 + 3) × 4?
    3. What is 36 − 9 ÷ 3 − 2?
    4. Which inequality says the same as 5 > x?
    5. On a number line, which inequality is shown with a filled circle?
    6. Which list is in order, smallest first?
    7. Which of these lengths is the longest?
    8. Where do brackets go to make 6 × 2 + 1 = 18 correct?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 3 (Calculations and order). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.