Topic 1 Number (9260): Revision Notes

Condensed for the final weeks. For the full explanation and worked examples, use the Topic 1 Number study guide. All content below is assessable on the Extension tier paper across every one of Topic 1’s three sub-topics.

The three sub-topics

Sub-topic	Core skill
1.1 Structure and calculation	Four operations, correct order of operations, indices and surds
1.2 Fractions, decimals, percentages	Converting fluently between all three forms, compound percentage change
1.3 Ratio and proportion	Direct vs. inverse proportion

All content is assessable on the Extension tier. Number is the most self-contained of the topics in this specification, in the sense that it draws on very little prior content from elsewhere — but it is also one of the most heavily interconnected going forward, since algebra, geometry and statistics questions throughout the rest of the syllabus routinely embed a Number sub-skill (a percentage change, an index manipulation, a ratio) inside a larger multi-topic problem.

Percentage multipliers — memorise these

increase x%  ->  x (1 + x/100)
decrease x%  ->  x (1 - x/100)
n years compound growth at r%  ->  x (1 + r/100)^n
reverse percentage -> DIVIDE by the multiplier, don't multiply

Prime factorisation → HCF/LCM

Write each number as a product of primes. HCF = lowest power of each shared prime. LCM = highest power of every prime seen. Check: HCF × LCM = product of the two original numbers — this relationship only holds for exactly two numbers, so don’t rely on it as a shortcut when a question gives three or more values.

Direct vs. inverse proportion

• Direct: y = kx (both increase/decrease together).
• Inverse: y = k/x (one increases as the other decreases).
• Always find k from a given pair first, then apply it.

Index laws

Multiply → add indices. Divide → subtract indices. Power of a power → multiply indices. Anything to the power 0 = 1. Negative index → reciprocal. Fractional index → root. A fractional index like x^(a/b) combines two of these rules: raise to the power a, then take the bth root (or vice versa) — practise applying both operations, in either order, since either sequence gives the same result but candidates often freeze when the fraction isn’t in the “obvious” 1/n form.

Bounds

A value to a given accuracy lies within an interval (e.g. 4.6 kg to 1 d.p. → 4.55 ≤ m < 4.65). Greatest sum: both upper bounds. Greatest difference: upper − lower. Greatest quotient: upper ÷ lower. The same logic in reverse gives the smallest sum (both lower bounds), smallest difference (lower − upper, i.e. the smallest possible gap), and smallest quotient (lower ÷ upper).

Compound interest and repeated percentage change

Compound growth/decay uses a single multiplier raised to a power, not repeated separate calculations: an amount growing by 5% a year for 3 years is x × 1.05³, not x × 1.05 × 3. This is one of the most consistently tested distinctions in Topic 1, since the two calculations produce noticeably different answers and a candidate who confuses them will not necessarily notice the error without checking the method itself. For decay (e.g. depreciation), the multiplier is (1 − r/100) raised to the same power.

Standard form

A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. Adding or subtracting numbers in standard form requires matching the powers of 10 first (converting one number so both share the same exponent) before combining the A values; multiplying or dividing combines the A values directly and adds or subtracts the powers of 10 separately.

Surds

A surd is an irrational root left in root form rather than evaluated as a decimal (e.g. √3, not 1.732…). Simplifying a surd means extracting the largest perfect-square factor: √12 = √(4×3) = 2√3. Rationalising a denominator (removing a surd from underneath a fraction) multiplies both numerator and denominator by the surd itself, or by its conjugate when the denominator is a sum or difference involving a surd.

Exam traps

• Reverse percentage: multiplying by the discount multiplier instead of dividing by it.
• Treating 1 as a prime number (it isn’t).
• Standard form written with A outside 1 ≤ A < 10 (e.g. 15 × 10⁴ instead of 1.5 × 10⁵).
• Using the upper bound of the denominator when finding the greatest possible quotient.
• Applying a repeated-percentage multiplier as a straight multiplication instead of raising it to a power.
• Leaving a surd un-simplified, or a denominator un-rationalised, when the question specifically asks for an answer “in the form a√b.”

Self-test

1. A price after a 15% discount is $68. What was the original price (as a calculation)?
2. Find the HCF and LCM of 60 and 84 using prime factorisation.
3. Is “y decreases as x increases, and y = k/x” direct or inverse proportion?
4. A length is 8.3 cm to 1 d.p. State the bounds.

Answers: 1. $68 ÷ 0.85 = $80. 2. 60 = 2²×3×5, 84 = 2²×3×7; HCF = 2²×3 = 12; LCM = 2²×3×5×7 = 420. 3. Inverse proportion. 4. 8.25 ≤ length < 8.35.

5. $2000 is invested at 4% compound interest for 3 years. Write the calculation for the final amount.
6. Simplify √50.

Answers (continued): 5. $2000 × 1.04³. 6. √50 = √(25×2) = 5√2.