Grade 9 Numbers, Operations and Relationships

Grade 9 numbers: integers, fractions, decimals, order of operations, rational and irrational numbers, ratio and rate, explained step by step with examples.

Grade 9 brings together everything you've learned about numbers: whole numbers, integers, fractions, decimals and percentages. You use them in calculations with several operations, and you learn to sort numbers into natural numbers, whole numbers, integers, rational numbers and irrational numbers.

Being quick and accurate with numbers matters because every later topic, from algebra in Grade 9 to finance and statistics in Grade 12, depends on it. This guide covers the rules you need, with worked examples and the mistakes to watch for.

Key concepts

  • Order of operations: brackets, exponents, multiplication and division, then addition and subtraction
  • Calculating with integers, including negative numbers
  • Adding, subtracting, multiplying and dividing fractions and decimals
  • The number system: natural, whole, integers, rational and irrational numbers
  • Ratio, rate and direct and indirect proportion
  • Estimating and rounding off

Worked examples

Calculate: −2 × (−3)² + 10 ÷ (−5)

Exponents first: (−3)² = 9. Then multiply and divide: −2 × 9 = −18 and 10 ÷ (−5) = −2. Finally add: −18 + (−2) = −20.

Which of these numbers is irrational: √7, √9, 0.25 or −3/4?

√7 can't be written as a fraction of two integers, so it is irrational. √9 = 3, 0.25 = 1/4 and −3/4 are all rational.

Calculate: 2⅓ × 3/14

Change the mixed number to an improper fraction: 2⅓ = 7/3. Then 7/3 × 3/14 = 21/42 = ½.

If 6 workers take 10 days to build a wall, how long would 4 workers take?

More workers means fewer days, so this is indirect (inverse) proportion: 6 × 10 = 60 worker-days, and 60 ÷ 4 = 15 days.

Common mistakes

  • Working from left to right instead of following the order of operations.
  • Writing −3² = 9 (it is −9; only (−3)² = 9).
  • Adding fractions by adding the tops and the bottoms.
  • Treating inverse proportion as direct proportion.

Exam tips

  • Underline the operation you will do first before you calculate.
  • Estimate the answer first so you can spot a calculator slip.
  • Show the improper fraction step when you work with mixed numbers.

Frequently asked questions

What is the difference between a rational and an irrational number?

A rational number can be written as a fraction a/b of two integers (b ≠ 0), such as 0.75 = 3/4. An irrational number can't, such as √2 or π; its decimal never ends or repeats.

Is zero an integer?

Yes. Zero is a whole number and an integer, and it is rational (0 = 0/1). It is not a natural number in the CAPS definition, which starts at 1.

Why does order of operations matter?

It makes sure everyone gets the same answer. Without it, 2 + 3 × 4 could be 20 or 14; the correct answer is 14.

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