🔢 Numbers, Ratio & Percentage
The Number strand covers prime factorisation, HCF/LCM, indices, standard form, ratio, proportion, percentage, rate and speed. These topics appear in both Paper 1 and Paper 2 and are fundamental to all other strands. Weakness in numbers will cripple your ability to handle algebra, geometry, and statistics. This chapter provides comprehensive coverage with detailed worked examples for every subtopic.
Prime Factorisation, HCF & LCM
Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers. This fundamental result is called the Fundamental Theorem of Arithmetic. We express the factorisation using index (power) notation for clarity and efficiency. For example, instead of writing 72 = 2 x 2 x 2 x 3 x 3, we write .
The primary reason we learn prime factorisation is to find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two or more numbers. The HCF is the largest number that divides evenly into all given numbers, while the LCM is the smallest number that all given numbers divide into evenly. These concepts have practical applications in problems involving repeated events, sharing equally, and finding common denominators.
To find HCF: Take the product of all common prime factors, each raised to the lowest power that appears in any factorisation.
To find LCM: Take the product of all prime factors, each raised to the highest power that appears in any factorisation.
🔢Prime Factorisation, HCF & LCM
Express numbers as products of prime factors using index notation. Find HCF by taking lowest powers of common primes; find LCM by taking highest powers of all primes.
📝Worked Example 1: Finding HCF and LCM
Find the HCF and LCM of 72 and 120.
📝Worked Example 2: HCF/LCM Word Problem
Two buses leave the same station at 8:00 am. Bus A returns every 18 minutes and Bus B returns every 24 minutes. When will they next be at the station together?
Find the HCF of 48 and 84.
Laws of Indices (Exponents)
The laws of indices are among the most important topics in E-Math. They appear in almost every Paper 1 and are also essential for algebraic manipulation in later chapters. You must be able to apply all seven laws fluently and confidently without a calculator. These laws allow you to simplify complex expressions involving powers, making them more manageable.
The key idea is that indices provide a shorthand for repeated multiplication. The expression means "multiply $a$ by itself $n$ times". The seven laws follow logically from this definition, but you should memorise them as rules for fast application in exams.
⚡Laws of Indices
Rules for simplifying expressions with exponents. These must be memorised for Paper 1 (non-calculator).
📝Worked Example 3: Simplifying Index Expressions
Simplify .
📝Worked Example 4: Negative and Zero Indices
Evaluate .
📝Worked Example 5: Fractional Indices
Evaluate and simplify .
Simplify:
Standard Form (Scientific Notation)
Standard form expresses a number as where and is an integer. This notation is used for very large numbers (like the distance to the sun: km) and very small numbers (like the mass of an electron: kg). The value of A must be at least 1 but strictly less than 10 — this is a common trap in exam questions.
To convert a number to standard form, move the decimal point until you have a number between 1 and 10. Count the number of places moved: if you moved the decimal left (large number), n is positive; if you moved it right (small number), n is negative.
🔬Standard Form
Express very large or very small numbers in standard form (scientific notation).
📝Worked Example 6: Standard Form Calculations
Express 0.00456 in standard form, then calculate .
📝Worked Example 7: Standard Form Division
Calculate , giving your answer in standard form.
What is 0.00456 in standard form?
Ratio & Proportion
A ratio compares two or more quantities of the same kind. Ratios are written in the form a : b (read as "a to b") and should always be simplified to their lowest terms. In direct proportion, as one quantity increases, the other increases at the same rate (y = kx). In inverse proportion, as one quantity increases, the other decreases (y = k/x). These relationships are tested frequently in word problems.
📝Worked Example 8: Ratio Problem
The ratio of boys to girls in a class is 3 : 5. If there are 40 students in total, find the number of boys and girls.
📝Worked Example 9: Direct & Inverse Proportion
y is inversely proportional to x. When x = 4, y = 6. Find (a) the equation, (b) y when x = 8, (c) x when y = 3.
Percentage, Simple & Compound Interest
Percentage problems are very common in both papers. The key concept that trips students up is reverse percentage — when you are given the final amount after a percentage change and need to find the original amount. Remember: if something increased by 20%, the new amount is 120% of the original. To find the original, divide by 1.2, do NOT subtract 20% from the new amount.
For interest calculations, simple interest is calculated only on the original principal amount, while compound interest is calculated on the principal plus any previously accumulated interest. Compound interest grows exponentially, which is why it is used in most real-world financial products.
💰Percentages & Interest
Percentage increase/decrease, simple interest, and compound interest formulas.
📝Worked Example 10: Reverse Percentage
After a 20% discount, a bag costs $64. Find the original price.
📝Worked Example 11: Percentage Increase
A town population increased from 45,000 to 52,200. Find the percentage increase.
📝Worked Example 12: Compound Interest
$5000 is invested at 4% per annum compound interest for 3 years. Find (a) the total amount and (b) the interest earned.
$5000 is invested at 4% compound interest for 3 years. What is the total amount?
Rate, Speed, Distance & Time
Speed, distance, and time problems require you to use the triangle relationship: Speed = Distance / Time. The most common error is calculating average speed incorrectly. Average speed is NOT the average of the individual speeds — it is the total distance divided by the total time.
🏎️Ratio, Rate & Speed
Ratio, proportion, rate, and speed-distance-time relationships.
📝Worked Example 13: Average Speed
A car travels 120 km at 60 km/h and then 80 km at 40 km/h. Find the average speed for the entire journey.
📝Worked Example 14: Unit Conversion in Speed
Convert 72 km/h to m/s.
📝Worked Example 14b: Multi-Leg Journey
A cyclist rides 20 km at 10 km/h, then rides 30 km at 15 km/h, then walks 5 km at 5 km/h. Find the average speed for the entire journey.
📝Worked Example 14c: Distance-Speed-Time with Stops
A bus leaves Town A at 08:00 and arrives at Town B (120 km away) at 10:30. It stops for 15 minutes at a rest stop along the way. Find the average speed (a) including the stop and (b) excluding the stop.
Number Properties and Surds
In Paper 1, you may be asked to simplify expressions involving surds (irrational square roots). The key operations are simplifying surds like , rationalising the denominator, and performing arithmetic with surds. A surd is “simplified” when the number under the square root has no perfect square factors.
📝Worked Example 14d: Simplifying Surds
Simplify (a) , (b) , (c) .
📝Worked Example 14e: Rationalising the Denominator
Express in the form where is a rational number.
Simplify .
⚠️Common Mistakes — Numbers & Arithmetic
Why: HCF (Highest Common Factor) is the largest factor common to both numbers. LCM (Lowest Common Multiple) is the smallest multiple common to both. Students often mix up HCF and LCM.
Why: When multiplying same bases, ADD the powers (not multiply). The rule is a^m x a^n = a^(m+n).
Why: For power of a power, MULTIPLY the indices. The rule is (a^m)^n = a^(mn). Students confuse this with the addition rule.
Why: Any non-zero number raised to the power of 0 equals 1, not 0. A common but costly error.
Why: A negative index means the reciprocal, not a negative number. a^(-n) = 1/a^n.
Why: For numbers less than 1, the power of 10 is negative. Moving the decimal point to the right gives a negative power.
Why: For reverse percentage: 80% of original = $80, so original = 80/0.80. Do not add 20% to the discounted price.
Why: Simple interest adds the same amount each year. Compound interest reinvests, so the amount grows exponentially.
Why: Average speed is NOT the average of two speeds. You must calculate total distance divided by total time.
Convert 20 m/s to km/h.
More on Proportion
Proportion questions come in two main forms. In direct proportion (), the graph is a straight line through the origin. When x doubles, y doubles. In inverse proportion (), the graph is a hyperbola. When x doubles, y halves. You may also encounter (y proportional to x squared) or .
📝Worked Example 15: Proportion with Square
y is directly proportional to x squared. When x = 3, y = 36. Find y when x = 5.
📝Worked Example 16: Finding the Original after Two Percentage Changes
A price was increased by 10% and then decreased by 10%. The final price is $198. Find the original price.
📝Worked Example 17: Simple Interest vs Compound Interest
Compare the interest earned on $10,000 at 5% per annum for 4 years under (a) simple interest and (b) compound interest.
is inversely proportional to . If when , find when .
Paper 1 Number Skills Checklist
Before sitting Paper 1, make sure you can do all of the following without a calculator:
- Convert between fractions, decimals, and percentages
- Apply all 7 laws of indices including fractional and negative indices
- Convert to and from standard form
- Perform arithmetic with standard form numbers
- Calculate HCF and LCM using prime factorisation
- Solve reverse percentage problems
- Convert between km/h and m/s
Number Patterns & Sequences
Number pattern questions test your ability to spot a rule and express it algebraically. The most common patterns are arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms). You may also encounter triangular numbers, square numbers, and more complex sequences where the differences themselves form a pattern.
The key to any pattern question is to calculate the first differences (difference between consecutive terms). If the first differences are constant, it is an arithmetic sequence and the nth term formula is linear. If the first differences are not constant, calculate the second differences. If the second differences are constant, the nth term is a quadratic expression.
📝Worked Example 18: Finding the nth Term of an Arithmetic Sequence
The first four terms of a sequence are 5, 11, 17, 23, ... Find (a) the 50th term and (b) the general formula for the nth term.
📝Worked Example 19: Quadratic Number Patterns
The first five terms of a sequence are: 2, 6, 12, 20, 30, ... Find the general term.
📝Worked Example 20: Pattern from a Diagram
Matchstick patterns: Pattern 1 uses 4 matchsticks (a square), Pattern 2 uses 7, Pattern 3 uses 10. How many matchsticks for Pattern n? Which pattern uses 301 matchsticks?
The nth term of a sequence is . What is ?
Estimation & Significant Figures
Estimation questions require you to round each number to 1 significant figure before performing the calculation mentally. This is a Paper 1 skill (no calculator). Significant figures questions also appear when you are asked to give a final answer "correct to 3 significant figures" — you must know how to count significant figures correctly, especially with numbers like 0.00204 (3 s.f.) or 4050 (3 s.f. with the trailing zero being ambiguous).
The rules for significant figures are: (1) All non-zero digits are significant, (2) Zeros between non-zero digits are significant, (3) Leading zeros are NOT significant, (4) Trailing zeros after a decimal point ARE significant, (5) Trailing zeros in whole numbers are ambiguous unless written in standard form.
📝Worked Example 21: Estimation by Rounding to 1 Significant Figure
Estimate the value of .
📝Worked Example 22: Significant Figures — Tricky Cases
State the number of significant figures in each: (a) 0.00304 (b) 50200 (c) 6.020 (d) 0.0750
Estimate: by rounding each to 1 s.f.
Fractions Without a Calculator
Paper 1 tests your ability to add, subtract, multiply, and divide fractions without a calculator. The most common error is forgetting to find a common denominator before adding or subtracting. For mixed numbers, convert to improper fractions first. Always simplify your final answer to its lowest terms.
📝Worked Example 23: Mixed Number Operations
Calculate , giving your answer as a fraction in its simplest form.
📝Worked Example 24: Multiplying and Dividing Fractions
Calculate .
📝Worked Example 25: Recurring Decimals to Fractions
Convert (0.272727...) to a fraction in its simplest form.
Convert 0.333... (0.3 recurring) to a fraction.
⚠️Common Mistakes — Numbers
Why: HCF (Highest Common Factor) is the largest factor common to both numbers. LCM (Lowest Common Multiple) is the smallest multiple common to both. Students often mix up HCF and LCM.
Why: When multiplying same bases, ADD the powers (not multiply). The rule is a^m x a^n = a^(m+n).
Why: For power of a power, MULTIPLY the indices. The rule is (a^m)^n = a^(mn). Students confuse this with the addition rule.
Why: Any non-zero number raised to the power of 0 equals 1, not 0. A common but costly error.