Formula Centre

Formula guides and key subject rules for every grade and subject, with worked examples and practice.

Addition Rule for Probability

Mathematics • Statistics and Probability • Grade 11

P(A or B) = P(A) + P(B) − P(A and B)
P(A)Probability of event A (—)
P(B)Probability of event B (—)

When to use it

Use it when two events can happen together and you must not double-count the overlap.

Worked example

P(A) = 0,5, P(B) = 0,4, P(A and B) = 0,2 → P(A or B) = 0,5 + 0,4 − 0,2 = 0,7.

Common mistakes

  • Leaving out the overlap term for non-mutually-exclusive events

Practice question

If P(A) = 0,45, P(B) = 0,3 and P(A and B) = 0,1, find P(A or B).

Compound Interest

Mathematics • Finance, Growth and Decay • Grade 12

A = P(1 + i)ⁿ
AAccumulated amount (R)
PPrincipal invested (R)
iInterest rate per period (decimal) (—)
nNumber of periods (periods)

When to use it

Use it for investments or loans where interest is earned on interest.

Worked example

R10 000 at 8% p.a. for 5 years → A = 10 000(1,08)⁵ = R14 693,28.

Common mistakes

  • Using 8 instead of 0,08 for i
  • Not converting to monthly periods when compounding monthly

Practice question

R25 000 is invested at 6% p.a. compounded monthly for 3 years. Calculate A.

Cosine Rule

Mathematics • Trigonometry • Grade 12

a² = b² + c² − 2bc·cosA
aSide opposite angle A (units)
b, cThe two known sides (units)
AIncluded angle (degrees)

When to use it

Use it in a non-right-angled triangle when you know two sides and the included angle, or all three sides.

Worked example

b = 8, c = 6, A = 60° → a² = 64 + 36 − 48 = 52, so a ≈ 7,21 units.

Common mistakes

  • Using it when the angle is not between the two known sides
  • Calculator in radian mode

Practice question

In △PQR, q = 10, r = 7 and P = 45°. Calculate p.

Derivative of a Power

Mathematics • Calculus • Grade 12

f(x) = axⁿ ⇒ f′(x) = naxⁿ⁻¹
aCoefficient (—)
nExponent (—)

When to use it

Use it to differentiate any polynomial term, then to find gradients, tangents and stationary points.

Worked example

f(x) = 4x³ ⇒ f′(x) = 12x².

Common mistakes

  • Forgetting that the derivative of a constant is 0
  • Subtracting 1 from the coefficient instead of the exponent

Practice question

Differentiate f(x) = 5x⁴ − 3x² + 7.

Grade 1 Mathematics essentials

Mathematics • Essential reference • Grade 1

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 10 Mathematics essentials

Mathematics • Essential reference • Grade 10

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 2 Mathematics essentials

Mathematics • Essential reference • Grade 2

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 3 Mathematics essentials

Mathematics • Essential reference • Grade 3

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 4 Mathematics essentials

Mathematics • Essential reference • Grade 4

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 5 Mathematics essentials

Mathematics • Essential reference • Grade 5

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 6 Mathematics essentials

Mathematics • Essential reference • Grade 6

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 7 Mathematics essentials

Mathematics • Essential reference • Grade 7

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 8 Mathematics essentials

Mathematics • Essential reference • Grade 8

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade 9 Mathematics essentials

Mathematics • Essential reference • Grade 9

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Grade R Mathematics essentials

Mathematics • Essential reference • Grade R

Core relationship: result = known values combined with the correct operation
resultThe value being calculated (Depends on the question)

When to use it

Choose the operation or formula that matches the problem, substitute known values, then check the answer.

Worked example

Identify what is known, select the matching rule, calculate step by step, and verify the result.

Common mistakes

  • Using an operation without checking what the question asks.

Practice question

Write the correct rule for a problem from this grade, then solve it.

Quadratic Formula

Mathematics • Algebra and Equations • Grade 12

x = [−b ± √(b² − 4ac)] / 2a
aCoefficient of x² (—)
bCoefficient of x (—)
cConstant term (—)

When to use it

Use it to solve any quadratic equation ax² + bx + c = 0, especially when the expression does not factorise.

Worked example

Solve 2x² + 3x − 2 = 0. x = [−3 ± √(9 + 16)]/4 = (−3 ± 5)/4, so x = 0,5 or x = −2.

Common mistakes

  • Forgetting the ± and losing a root
  • Dividing only part of the numerator by 2a

Practice question

Solve 3x² − 5x − 2 = 0 using the quadratic formula.