Fractions, Decimals & Percentages
Adding & Subtracting Fractions
To add or subtract fractions, you need a common denominator.
Worked Example
Calculate 23 + 14
Step 1: Find common denominator: LCD of 3 and 4 = 12
Step 2: 812 + 312 = 1112
Multiplying & Dividing Fractions
Multiply: Multiply the numerators together and the denominators together.
Divide: Flip the second fraction (take its reciprocal) and multiply.
Converting Recurring Decimals to Fractions
Worked Example
Convert 0.272727... to a fraction.
Step 1: Let x = 0.272727...
Step 2: 100x = 27.272727...
Step 3: 100x − x = 27
Step 4: 99x = 27, so x = 2799 = 311
Percentage Multipliers
To increase by a percentage, multiply by 1 + (% as decimal). To decrease, multiply by 1 − (% as decimal).
Reverse percentage: To find the original amount, divide by the multiplier.
Key Facts
- To convert a fraction to a decimal, divide the numerator by the denominator
- To convert a decimal to a percentage, multiply by 100
- Always simplify your fractions at the end
Exam Tip
For percentage problems, identify the multiplier first. It makes compound percentage questions much faster!
Factors, Multiples & Primes
Prime Factor Decomposition
Every integer greater than 1 can be written as a product of prime numbers. Use a factor tree or repeated division.
Worked Example
Write 120 as a product of its prime factors.
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 is prime. So 120 = 2³ × 3 × 5
Finding HCF and LCM
Using prime factor decomposition:
HCF: Multiply the common prime factors (use the lowest power of each).
LCM: Multiply all prime factors (use the highest power of each).
Worked Example
Find the HCF and LCM of 24 and 36.
24 = 2³ × 3
36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72
Key Facts
- 1 is NOT a prime number
- 2 is the only even prime number
- HCF × LCM = product of the two numbers
Indices (Powers)
Laws of Indices
am ÷ an = am−n
(am)n = amn
Special Index Rules
Worked Example
Evaluate 82/3
Step 1: Find the cube root: 3√8 = 2
Step 2: Square the result: 2² = 4
So 82/3 = 4
Exam Tip
For fractional indices, always do the ROOT first, then the POWER. This keeps the numbers smaller and easier to work with.
Standard Form
Writing Numbers in Standard Form
Standard form is written as A × 10n where 1 ≤ A < 10.
Small numbers: 0.0032 = 3.2 × 10−3
Worked Example
Calculate (3 × 104) × (2 × 105)
Step 1: Multiply the numbers: 3 × 2 = 6
Step 2: Add the powers: 104 × 105 = 109
Answer: 6 × 109
Common Mistake
If A ends up ≥ 10 or < 1 after calculation, you must adjust. E.g. 15 × 10³ = 1.5 × 10&sup4;
Surds
Simplifying Surds
Look for the largest square number factor inside the root.
Worked Example
Simplify √48
√48 = √(16 × 3)
= √16 × √3
= 4√3
Rationalising the Denominator
Multiply the top and bottom by the surd in the denominator.
Worked Example
Rationalise 5√3
= 5√3 × √3√3
= 5√33
Key Facts
- √a × √a = a
- (√a + √b)(√a − √b) = a − b (difference of two squares)
Bounds & Error Intervals
Upper and Lower Bounds
When a measurement is rounded, the true value lies within an error interval.
Lower bound ≤ x < Upper bound
Worked Example
A length is 5.4 cm to 1 decimal place. Find the error interval.
Degree of accuracy = 0.1, so error = 0.05
Lower bound = 5.4 − 0.05 = 5.35
Upper bound = 5.4 + 0.05 = 5.45
Error interval: 5.35 ≤ x < 5.45
Calculations with Bounds
For the maximum value of a calculation:
- Addition/Multiplication: use upper bounds
- Subtraction/Division: upper bound of first, lower bound of second
For the minimum value, do the opposite.
Exam Tip
Think about what makes the answer bigger or smaller. For maximum area, use both upper bounds. For maximum speed = distance/time, use upper distance and lower time.
Number Flashcards
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Number Quiz
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Number - Mock Exam Questions
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