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

The specification’s own content references use an N-prefix (N1-N20) running through this topic area, not section-heading numbering:

Sub-topic	Content refs	Core skill
3.1.1 Structure and calculation	N1-N11	Four operations, correct order of operations, indices and surds
3.1.2 Fractions, decimals, percentages	N12-N15	Converting fluently between all three forms, compound percentage change
3.1.3 Ratio and proportion	N16-N20	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 (N11) — Extension tier only

Core candidates round to a given accuracy but are not required to calculate or use 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 (N15)

The compound-interest formula below, and reverse percentage problems, are Extension tier only; Core candidates solve compound-interest and simple percentage-change problems without being required to use or quote the formula.

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 (N7) — Extension tier only

Core candidates calculate exactly with fractions instead; surds are not required at Core.

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.