Essential Number Properties and Percentage Calculations

Posted on Sep 27, 2026 in Mathematics

1. Prime Factorisation, HCF, and LCM

Prime Factorisation:
- Breaking a number down into a product of its prime factors (usually using factor trees).
- Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
Highest Common Factor (HCF):
- The largest factor that divides two or more numbers.
- Method (Venn Diagram): Multiply the numbers found in the intersection (overlap) of the prime factor sets.
Lowest Common Multiple (LCM):
- The smallest multiple shared by two or more numbers.
- Method (Venn Diagram): Multiply all numbers appearing inside the union of both sets.
- Quick Check: HCF × LCM = Number A × Number B.

Equivalent Fractions:
- Multiply or divide both the numerator and denominator by the exact same non-zero number.
- Example: 2/5 = (2 × 4) / (5 × 4) = 8/20.
Converting Mixed Numbers to Improper Fractions:
- Rule: Multiply the whole number by the denominator, add the numerator, and place it over the original denominator.
- Formula: A (b/c) = [(A × c) + b] / c.
- Example: 3 (2/5) = [(3 × 5) + 2] / 5 = 17/5.
Converting Improper Fractions to Mixed Numbers:
- Divide the top by the bottom: The whole quotient becomes the integer, and the remainder stays over the denominator.
- Example: 22/7 = 3 with remainder 1 = 3 (1/7).
Reciprocal:
- The reciprocal of a number x is 1/x (turn the fraction upside down).
- A number multiplied by its reciprocal always equals 1.
- Examples:
  - Reciprocal of 4 is 1/4.
  - Reciprocal of 2/3 is 3/2.
  - Reciprocal of 1 (1/2) → Convert to 3/2 first → Reciprocal is 2/3.

Rational Numbers:
- Any number that can be expressed as a fraction a/b where a and b are integers and b ≠ 0.
- Includes: Integers (-4, 7), terminating decimals (0.75 = 3/4), and recurring decimals (0.333… = 1/3).
Irrational Numbers:
- Non-terminating, non-repeating decimals.

Let x equal the recurring decimal: x = 0.777…
Multiply by a power of 10 (10, 100, 1000) to shift one full repeating block past the decimal: 10x = 7.777…
Subtract the equations to cancel the repeating part: 10x – x = 7.777… – 0.777… → 9x = 7.
Solve for x: x = 7/9.

Example with 2 repeating digits: x = 0.4545… → 100x = 45.4545… → 99x = 45 → x = 45/99 = 5/11.

Step 1: Let x = 0.6787878…
Step 2 (Move the 6): Multiply by 10 → 10x = 6.787878…
Step 3 (Move the 6 and one 78 block): Multiply original x by 1000 → 1000x = 678.787878…
Step 4 (Subtract): 1000x – 10x = 678.787878… – 6.787878… → 990x = 672
Step 5 (Solve and simplify): x = 672 / 990. Divide top and bottom by 6: x = 112 / 165

Writing an Amount as a % of Another:
- Formula: (Amount / Total) × 100%
- Example: Score 36 out of 80 → (36 / 80) × 100 = 0.45 × 100 = 45%.
Percentage Increase:
- Multiplier Method: New Value = Original × (1 + % / 100).
- Example: Increase £80 by 15% → Multiplier = 1.15 → 80 × 1.15 = £92.
- Percentage Change Formula: % Increase = (Increase / Original) × 100%.
Percentage Profit / Loss:
- Profit = Selling Price – Cost Price
- Loss = Cost Price – Selling Price
- Formula: % Profit or Loss = (Profit or Loss / Cost Price) × 100%
- Crucial: Always divide by the Original / Cost Price, never the selling price.
Reverse Percentages (Finding the Original Amount):
- Formula: Original = Final Value / Multiplier
- If an item increased by 20% to £72: Multiplier = 1.20 → Original = 72 / 1.20 = £60.
- If an item is reduced by 15% to £170: Multiplier = 0.85 → Original = 170 / 0.85 = £200.
- Crucial: Never find the percentage of the new value and add/subtract it.

Concept	Formula / Key Method	Note
Writing A as % of B	(A / B) × 100	Units of A and B must match
% Profit or Loss	(Profit or Loss / Cost Price) × 100	Always divide by original cost
Percentage Multiplier	1 + (% / 100) for increase; 1 – (% / 100) for decrease	Use decimals (e.g. +12% = 1.12)
Reverse %	Original = New Value / Multiplier	Never take % of the new value
Reciprocal of a/b	b/a	Product of number and reciprocal is 1
Simple Interest	Interest = (P × R × T) / 100	Adds a fixed amount each year
Compound Interest	Amount = P × (1 + r)^t	Multiplier applied t times
Recurring Decimal to Fraction	Set equal to x; multiply by 10/100/1000; subtract	Cancels infinite tail

Simple Interest: Interest calculated only on the starting amount.
- Interest: I = (P × R × T) / 100
- Total Amount: Principal + Interest
- Example: £500 at 4% for 3 yrs → I = (500 × 4 × 3) / 100 = £60 → Total = £560
Compound Interest: Interest calculated on the accumulated balance.
- Total Amount: A = P × (1 + r)^t
- Interest Earned: Amount – Principal
- (P = principal, r = decimal rate, t = years)
- Example: £2,000 at 3% for 4 yrs → A = 2000 × (1.03)&sup4; = £2,251.02