Mathematics Cheat Sheet: Essential Rules, Formulas, and Examples

Mathematics Term 3 Reference

Chapter 2

Divisibility Tests

• 1: All numbers are divisible by 1.
• 2: All even numbers are divisible by 2. Last digit must be 0, 2, 4, 6, or 8.
• 3: The sum of the digits must be divisible by 3.
• 4: The number formed from the last two digits must be divisible by 4.
• 5: The last digit must be 0 or 5.
• 6: Must pass the divisibility tests for both 2 and 3.
• 7: (There is no simple test for divisibility by 7).
• 8: The number formed from the last three digits must be divisible by 8.
• 9: The sum of the digits must be divisible by 9.
• 10: The last digit must be 0.

Terms for Division

• Dividend: The starting number; the total; the amount you have.
• Divisor: The number doing the dividing; the number of groups.
• Quotient: The number of times the divisor goes into the dividend; also known as 'the answer' to a division calculation.
• Remainder: The number left over; the number remaining (sometimes written as 'rem.').

Prime Numbers and Index Laws

A prime number is defined as a positive whole number that has exactly two distinct factors: itself and 1. The numbers 0 and 1 are neither prime nor composite numbers. The number 2 is prime and is the only even prime number. (+ and × are related; - and ÷ are related)

• Multiplication Law: When multiplying terms with the same base, you add the powers.
  • Example: a³ × a² = a³⁺² = a⁵
• Division Law: When dividing terms with the same base, you subtract the powers.
  • Example: a³ ÷ a² = a³⁻² = a¹
• Power of a Power Law: When an index is raised to another power, you multiply the powers together.
  • Example: (a³)² = a³ˣ² = a⁶
• Zero Index: Any non-zero number or variable raised to the power of zero is always equal to 1.
  • Example: a⁰ = 1
• Power of a Product Law: A power outside a bracket applies to every term inside the bracket.
  • Example: (ab)² = a²b²

Squares, Cubes, and Roots

To produce a square number, you multiply the number by itself. All square numbers written in index form will have a power of 2. Example: 2 × 2 = 2² = 4. Finding the square root of a number is the inverse of squaring a number. Example: 4 × 4 = 16, therefore √16 = 4.

Cubes are formed by multiplying a number by itself 3 times. All cube numbers written in index form will have a power of 3. Example: 2 × 2 × 2 = 2³ = 8. Cube roots are the inverse of cubing a number. The cube root symbol includes a small three on the left side. Example: ³√64 = 4.

Fractions: Terms and Principles

• Equivalent Fractions: Fractions that mark the same point on a number line. Example: 1/2 = 2/4 = 4/8
• Simplifying Fractions: Writing a fraction in its simplest form, where the numerator and denominator share no common factors other than 1. Divide both by their Highest Common Factor (HCF). Example: 9/12 = (9 ÷ 3) / (12 ÷ 3) = 3/4
• Ordering Fractions: To determine which fractions are larger or smaller, convert them to equivalent fractions with a Lowest Common Denominator (LCD). Example: 7/12 < 5/8 (since 14/24 < 15/24)
• Improper Fractions: Generally, improper fractions should be converted to mixed numerals unless directed otherwise. Example: 11/3 = 3 2/3

Adding and Subtracting Fractions

Fractions with the same denominator can be added directly. Example: 3/10 + 2/10 = 5/10 = 1/2

Fractions with different denominators require finding a common denominator first. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12

Subtracting fractions follows the same principle by converting to equivalent fractions with an LCD. Example: 10/12 - 1/3 = 10/12 - 4/12 = 6/12 = 1/2. A fail-safe method for subtracting mixed numerals is to convert them to improper fractions first. Example: 7 1/2 - 2 3/4 = 15/2 - 11/4 = 30/4 - 11/4 = 19/4 = 4 3/4.

Multiplying and Dividing Fractions: What does 1/3 × 2/3 equal? Remember: Addition and subtraction use cross-multiplication/LCD methods, whereas multiplication is performed straight across numerator-by-numerator and denominator-by-denominator. For division, use the reciprocal rule: KEEP, CHANGE, FLIP (Keep the first fraction, Change to multiplication, Flip the second fraction).

If possible, simplify or cancel factors vertically or diagonally before multiplying. Note: Cancelling can never be done horizontally.

Mixed numerals MUST be converted to improper fractions before multiplying or dividing.

Decimals: Rounding Rules

Rounding involves approximating a decimal number to fewer decimal places:

• Identify the target decimal place (e.g., 2 decimal places).
• Look at the critical digit immediately to the right. If it is 0–4, round DOWN; if it is 5–9, round UP. Use the approximation symbol (≈) when writing rounded answers.

Decimals: Addition and Subtraction

When adding or subtracting decimal numbers, align the decimal points vertically and use standard formal algorithms.

Decimals: Multiplication and Division

Multiplying by powers of 10 (10, 100, 1000...): Move the decimal point to the right by as many places as there are zeros in the multiplier. Example: 5.9347 × 100 = 593.47

Dividing by powers of 10 (10, 100, 1000...): Move the decimal point to the left by as many places as there are zeros in the divisor. Example: 12364 ÷ 10 = 1236.4

Multiplying Decimals: Multiply the numbers ignoring decimal points first. Then, count the total number of decimal places in the original numbers and place the decimal point that many places from the right in the product.

Estimations can be made by rounding numbers to the nearest whole number first. Example: 2.93 × 5.17 ≈ 3 × 5 = 15

Dividing Decimals: When dividing by decimals, multiply both the dividend and divisor by powers of 10 until the divisor becomes a whole number.

When dividing multiples of 10, perform the basic division first, then move the decimal point to the left as needed. Example: 4.5 ÷ 30 = (4.5 ÷ 3) ÷ 10 = 1.5 ÷ 10 = 0.15

Converting simple decimals to fractions: 0.75 = 75/100 = 3/4; 0.25 = 1/4.