Types of Number
Integers · Primes · HCF & LCM · Rational & Irrational · Ordering
Types of Number
Natural (ℕ): 1, 2, 3, 4, … (positive integers)
Integers (ℤ): …, −2, −1, 0, 1, 2, … (includes negatives)
Rational (ℚ): can be written as \(\frac{p}{q}\) where \(p, q\) are integers. Decimals terminate or recur.
Irrational: cannot be written as a fraction. e.g. \(\sqrt{2},\; \pi,\; e\)
Real (ℝ): all rational and irrational numbers
Prime Numbers
A prime has exactly two factors: 1 and itself. 1 is not prime.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47…
Prime factorisation
Express any integer as a product of primes. e.g. \(60 = 2^2 \times 3 \times 5\)
HCF & LCM
HCF
Highest Common Factor — largest number dividing both. Use prime factorisation: take minimum powers of common primes.
LCM
Lowest Common Multiple — smallest number both divide into. Take maximum powers of all primes.
Example: 12 = 2²×3, 18 = 2×3²
HCF = 2×3 = 6
LCM = 2²×3² = 36
HCF = 2×3 = 6
LCM = 2²×3² = 36
Rounding & Bounds
Sig figs
Count from the first non-zero digit. 0.00472 to 2 s.f. = 0.0047
Dec places
Count from the decimal point. 3.1416 to 2 d.p. = 3.14
Rules
5 and above → round up. Below 5 → round down.
Upper & Lower Bounds
When \(x = 6.3\) cm (1 d.p.):
Lower bound = \(6.25\) cm
Upper bound = \(6.35\) cm
Interpretation: \(6.25 \leq x < 6.35\)
Lower bound = \(6.25\) cm
Upper bound = \(6.35\) cm
Interpretation: \(6.25 \leq x < 6.35\)
Trap: Upper bound is just below 6.35, not 6.35 itself (it's the value that would round up to 6.4).
Fractions, Decimals & Percentages
Converting between forms · Percentage change · Reverse percentage · Ratio
Fraction Operations
Add/Sub
Use LCM of denominators: \(\dfrac{1}{3}+\dfrac{1}{4}=\dfrac{4}{12}+\dfrac{3}{12}=\dfrac{7}{12}\)
Multiply
Top × top, bottom × bottom: \(\dfrac{2}{3}\times\dfrac{4}{5}=\dfrac{8}{15}\)
Divide
Flip and multiply: \(\dfrac{2}{3}\div\dfrac{4}{5}=\dfrac{2}{3}\times\dfrac{5}{4}=\dfrac{10}{12}=\dfrac{5}{6}\)
Simplify first — cancel common factors before multiplying to keep numbers smaller.
Percentage Calculations
% of amount
\(p\%\) of \(x = \dfrac{p}{100}\times x\)
% change
\(\dfrac{\text{change}}{\text{original}}\times100\%\)
Multiplier
Increase by 20% → multiply by 1.20. Decrease by 15% → multiply by 0.85.
Reverse %
After 20% increase, price = $360. Original = \(\dfrac{360}{1.2} = \$300\)
Reverse % trap: DO NOT subtract 20% from $360. Always divide by the multiplier.
Ratio & Proportion
Simplify
Write ratio in lowest terms: \(15:25 = 3:5\)
Sharing
Share £48 in ratio 3:5. Total parts = 8. One part = £6. Shares: £18 and £30.
Direct prop.
\(y\propto x\): \(y=kx\). Find \(k\) from one known pair, then use for others.
Inverse prop.
\(y\propto\frac{1}{x}\): \(y=\frac{k}{x}\). As \(x\) doubles, \(y\) halves.
Compound Interest vs Simple Interest
Simple: \(I = PRT\) (interest on original only)
Compound: \(A = P\left(1+\dfrac{r}{100}\right)^n\)
Principal £2000, rate 5%, 3 years:
Simple: \(I = 2000\times0.05\times3 = £300\)
Compound: \(A = 2000(1.05)^3 \approx £2315.25\)
Compound: \(A = P\left(1+\dfrac{r}{100}\right)^n\)
Principal £2000, rate 5%, 3 years:
Simple: \(I = 2000\times0.05\times3 = £300\)
Compound: \(A = 2000(1.05)^3 \approx £2315.25\)
Converting Between Forms
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅓ | \(0.\overline{3}\) | 33.3% |
| ⅛ | 0.125 | 12.5% |
| ⅕ | 0.2 | 20% |
Indices & Standard Form
Index laws · Negative & fractional indices · Standard form calculations
Index Laws — Summary
\(a^m \times a^n = a^{m+n}\)
\(\dfrac{a^m}{a^n} = a^{m-n}\)
\((a^m)^n = a^{mn}\)
\(a^0 = 1\)
\(a^{-n} = \dfrac{1}{a^n}\)
\(a^{p/q} = \left(\sqrt[q]{a}\right)^p\)
Fractional index rule: The denominator is the root, the numerator is the power. \(8^{2/3} = (\sqrt[3]{8})^2 = 4\). Always take the root first — then power.
Standard Form (Scientific Notation)
\(A \times 10^n\) where \(1 \leq A < 10\) and \(n\) is an integer
Large num.
\(4\,500\,000 = 4.5 \times 10^6\)
Small num.
\(0.0000032 = 3.2 \times 10^{-6}\)
Add/Sub
Convert to same power of 10 first, then add/subtract A values.
Multiply
Multiply A values, add exponents: \((3\times10^4)\times(2\times10^3)=6\times10^7\)
Divide
Divide A values, subtract exponents: \(\dfrac{6\times10^8}{2\times10^3}=3\times10^5\)
Worked Examples
Evaluate \(27^{-2/3}\):
\(\sqrt[3]{27}=3 \Rightarrow 3^2=9 \Rightarrow 27^{2/3}=9\)
So \(27^{-2/3}=\dfrac{1}{9}\)
Solve \(2^{3x+1}=8^{x-2}\):
\(8=2^3\Rightarrow 2^{3x+1}=2^{3(x-2)}\)
\(3x+1=3x-6\) — contradiction! No solution.
Estimate \(\dfrac{612\times 0.0049}{0.238}\):
\(\approx\dfrac{600\times0.005}{0.25}=\dfrac{3}{0.25}=12\)
\(\sqrt[3]{27}=3 \Rightarrow 3^2=9 \Rightarrow 27^{2/3}=9\)
So \(27^{-2/3}=\dfrac{1}{9}\)
Solve \(2^{3x+1}=8^{x-2}\):
\(8=2^3\Rightarrow 2^{3x+1}=2^{3(x-2)}\)
\(3x+1=3x-6\) — contradiction! No solution.
Estimate \(\dfrac{612\times 0.0049}{0.238}\):
\(\approx\dfrac{600\times0.005}{0.25}=\dfrac{3}{0.25}=12\)
Surds
Simplify
\(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\)
Add/Sub
\(3\sqrt{5}+2\sqrt{5}=5\sqrt{5}\) (like terms only)
Multiply
\(\sqrt{3}\times\sqrt{12}=\sqrt{36}=6\)
Rationalise
\(\dfrac{4}{\sqrt{3}}=\dfrac{4\sqrt{3}}{3}\) — multiply by \(\dfrac{\sqrt{3}}{\sqrt{3}}\)
Conj. denom.
\(\dfrac{1}{3-\sqrt{2}}=\dfrac{3+\sqrt{2}}{(3)^2-(\sqrt{2})^2}=\dfrac{3+\sqrt{2}}{7}\)
Quick Surd Rules
\(\sqrt{a}\times\sqrt{a} = a\)
\(\sqrt{ab} = \sqrt{a}\times\sqrt{b}\)
\(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\)
\((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a - b\)
\(\sqrt{ab} = \sqrt{a}\times\sqrt{b}\)
\(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\)
\((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a - b\)
Practice Questions
3 Easy · 3 Medium · 3 Hard — attempt first, then reveal full solution
Easy 1
[2]
Find the HCF and LCM of 24 and 36.
STEP-BY-STEP SOLUTION
Factorise
\(24 = 2^3 \times 3\) and \(36 = 2^2 \times 3^2\)
HCF
Take minimum powers of common primes: \(2^2 \times 3 = \boldsymbol{12}\)
LCM
Take maximum powers of all primes: \(2^3 \times 3^2 = \boldsymbol{72}\)
Check: \(12 \times 72 = 864 = 24 \times 36\) ✓ (HCF × LCM = product of two numbers)
Easy 2
[2]
Write \(0.000\,058\,3\) in standard form, and write \(4.07\times10^5\) as an ordinary number.
STEP-BY-STEP SOLUTION
Small → std
Move decimal 5 places right to get 5.83. \(\boldsymbol{5.83\times10^{-5}}\)
Std → ordinary
Move decimal 5 places right: \(\boldsymbol{407\,000}\)
Easy 3
[3]
Without a calculator, evaluate: (a) \(16^{3/4}\) (b) \(125^{-2/3}\)
STEP-BY-STEP SOLUTION
(a) Root first
\(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = \boldsymbol{8}\)
(b) Root first
\(125^{2/3} = (\sqrt[3]{125})^2 = 5^2 = 25\)
(b) Negative
\(125^{-2/3} = \dfrac{1}{25} = \boldsymbol{0.04}\)
Medium 1
[4]
A jacket costs £85 after a 15% discount. Find the original price. Also, if the shop then applies a 10% increase to the discounted price, is the result more than, less than, or equal to the original price?
STEP-BY-STEP — REVERSE PERCENTAGE
Multiplier
15% off → multiply by 0.85. So discounted = original × 0.85 = £85.
Original
Original = \(\dfrac{85}{0.85} = \boldsymbol{£100}\)
10% increase
£85 × 1.10 = £93.50
Compare
£93.50 < £100 (original). Net effect: −15% then +10% = ×0.85×1.10 = ×0.935 = 6.5% reduction overall.
Medium 2
[4]
A length is measured as \(x = 12.4\) cm, rounded to 1 decimal place. Calculate the upper and lower bounds of the perimeter of a square with side \(x\), and state the perimeter to an appropriate degree of accuracy.
STEP-BY-STEP — BOUNDS
Bounds of x
Lower: \(12.35\) cm Upper: \(12.45\) cm
Perimeter LB
\(4 \times 12.35 = \boldsymbol{49.4}\) cm
Perimeter UB
\(4 \times 12.45 = \boldsymbol{49.8}\) cm
Accuracy
Both bounds round to \(49.6\) cm (1 d.p.), so perimeter = \(\boldsymbol{49.6}\) cm (1 d.p.)
Medium 3
[3]
Simplify \(\dfrac{(3.2\times10^7)\times(5\times10^{-3})}{4\times10^{-4}}\). Give your answer in standard form.
STEP-BY-STEP — STANDARD FORM CALCULATION
Numerator
\(3.2\times5=16\) and \(10^7\times10^{-3}=10^4\). Numerator: \(16\times10^4\)
Divide
\(\dfrac{16\times10^4}{4\times10^{-4}}=4\times10^{4-(-4)}=4\times10^8\)
Answer
\(\boldsymbol{4\times10^8}\)
Hard 1
[5]
Prove that \(0.\overline{27}\) is rational by expressing it as a fraction in lowest terms. Hence show that the sum \(0.\overline{27} + 0.\overline{3}\) is a rational number and find it.
STEP-BY-STEP — RECURRING DECIMALS
Let x
\(x = 0.\overline{27} = 0.272727\ldots\)
100x
\(100x = 27.272727\ldots\)
Subtract
\(99x = 27 \Rightarrow x = \dfrac{27}{99} = \dfrac{3}{11}\)
Add
\(0.\overline{3} = \dfrac{1}{3}\). Sum: \(\dfrac{3}{11}+\dfrac{1}{3}=\dfrac{9}{33}+\dfrac{11}{33}=\boldsymbol{\dfrac{20}{33}}\)
Rational + Rational = Rational. \(\frac{20}{33}\) is in lowest terms (no common factor of 20 and 33). ✓
Hard 2
[5]
Simplify \(\dfrac{\sqrt{75}-\sqrt{48}}{\sqrt{3}}\). Hence evaluate \(\left(\dfrac{\sqrt{75}-\sqrt{48}}{\sqrt{3}}\right)^3\) without a calculator.
STEP-BY-STEP — SURDS
Simplify surds
\(\sqrt{75}=5\sqrt{3}\) and \(\sqrt{48}=4\sqrt{3}\)
Numerator
\(5\sqrt{3}-4\sqrt{3}=\sqrt{3}\)
Simplify
\(\dfrac{\sqrt{3}}{\sqrt{3}}=\boldsymbol{1}\)
Cube
\(1^3=\boldsymbol{1}\)
Hard 3
[4]
The population of a town decreases by 3% each year. In 2020 the population was 50,000.
(a) Find the population in 2025.
(b) Find the year when the population first falls below 40,000.
(a) Find the population in 2025.
(b) Find the year when the population first falls below 40,000.
STEP-BY-STEP — COMPOUND DECREASE
(a)
\(P = 50000 \times (0.97)^5 = 50000 \times 0.8587 \approx \boldsymbol{42\,934}\)
(b) Inequality
\(50000\times(0.97)^n < 40000 \Rightarrow (0.97)^n < 0.8\)
(b) Log
\(n\ln(0.97)<\ln(0.8) \Rightarrow n>\dfrac{\ln0.8}{\ln0.97}\approx\dfrac{-0.2231}{-0.03046}\approx7.32\)
(b) Year
\(n=8\) → year \(2020+8=\boldsymbol{2028}\)
⚠ Log inequality: dividing by ln(0.97) which is negative flips the inequality sign.
Past Year Paper Questions
Cambridge IGCSE 0580 style — multi-part, show all working
0580 Style
HCF, LCM & Primes
[9]
(a) Write 360 as a product of its prime factors. [2]
(b) Find the HCF and LCM of 360 and 504. [3]
(c) Three lights flash at intervals of 12, 18 and 30 seconds. They all flash together at 09:00. When do they next all flash together? [4]
(b) Find the HCF and LCM of 360 and 504. [3]
(c) Three lights flash at intervals of 12, 18 and 30 seconds. They all flash together at 09:00. When do they next all flash together? [4]
FULL WORKED SOLUTION
(a)
\(360 = 2^3 \times 3^2 \times 5\)
(b) Factorise
\(504 = 2^3 \times 3^2 \times 7\)
HCF = \(2^3\times3^2 = \boldsymbol{72}\)
LCM = \(2^3\times3^2\times5\times7 = \boldsymbol{2520}\)
HCF = \(2^3\times3^2 = \boldsymbol{72}\)
LCM = \(2^3\times3^2\times5\times7 = \boldsymbol{2520}\)
(c) LCM
\(12=2^2\times3,\; 18=2\times3^2,\; 30=2\times3\times5\)
LCM \(=2^2\times3^2\times5=180\) seconds = 3 minutes
Next flash: \(\boldsymbol{09{:}03{:}00}\)
LCM \(=2^2\times3^2\times5=180\) seconds = 3 minutes
Next flash: \(\boldsymbol{09{:}03{:}00}\)
0580 Style
Percentages & Compound Interest
[10]
(a) In a sale, all prices are reduced by 20%. A television costs $480 in the sale. Find the original price. [2]
(b) Arjun invests $5000 at a compound interest rate of 4.5% per year. After how many whole years will his investment first exceed $6000? Show your working. [4]
(c) A car depreciates by 12% each year. It was bought for $18,000. Find its value after 3 years, and as a percentage of the original price. [4]
(b) Arjun invests $5000 at a compound interest rate of 4.5% per year. After how many whole years will his investment first exceed $6000? Show your working. [4]
(c) A car depreciates by 12% each year. It was bought for $18,000. Find its value after 3 years, and as a percentage of the original price. [4]
FULL WORKED SOLUTION
(a)
Original × 0.80 = $480. Original = \(\frac{480}{0.80} = \boldsymbol{\$600}\)
(b)
\(5000(1.045)^n > 6000 \Rightarrow (1.045)^n > 1.2\)
Try \(n=4\): \(1.045^4=1.193<1.2\). Try \(n=5\): \(1.045^5=1.246>1.2\).
Answer: \(\boldsymbol{n=5}\) years.
Try \(n=4\): \(1.045^4=1.193<1.2\). Try \(n=5\): \(1.045^5=1.246>1.2\).
Answer: \(\boldsymbol{n=5}\) years.
(c)
\(V = 18000\times(0.88)^3=18000\times0.6815=\boldsymbol{\$12\\,267}\)
Percentage: \(\frac{12267}{18000}\times100\approx\boldsymbol{68.2\%}\) of original
Percentage: \(\frac{12267}{18000}\times100\approx\boldsymbol{68.2\%}\) of original
0580 Style
Indices, Surds & Standard Form
[9]
(a) Without using a calculator, evaluate \(\left(\dfrac{8}{27}\right)^{-2/3}\). [3]
(b) Simplify \(\left(\dfrac{2\sqrt{3}+\sqrt{2}}{\sqrt{2}}\right)^2\). [4]
(c) Calculate \(\dfrac{(3.6\times10^{-4})}{(1.2\times10^2)\times(5\times10^{-8})}\). Give your answer in standard form. [2]
(b) Simplify \(\left(\dfrac{2\sqrt{3}+\sqrt{2}}{\sqrt{2}}\right)^2\). [4]
(c) Calculate \(\dfrac{(3.6\times10^{-4})}{(1.2\times10^2)\times(5\times10^{-8})}\). Give your answer in standard form. [2]
FULL WORKED SOLUTION
(a)
\(\left(\frac{8}{27}\right)^{2/3} = \left(\sqrt[3]{\frac{8}{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\)
Negative exponent: \(\boldsymbol{\frac{9}{4} = 2.25}\)
Negative exponent: \(\boldsymbol{\frac{9}{4} = 2.25}\)
(b)
\(\dfrac{2\sqrt{3}+\sqrt{2}}{\sqrt{2}} = \dfrac{2\sqrt{3}}{\sqrt{2}}+1 = \sqrt{6}+1\)
Squaring: \((\sqrt{6}+1)^2=6+2\sqrt{6}+1=\boldsymbol{7+2\sqrt{6}}\)
Squaring: \((\sqrt{6}+1)^2=6+2\sqrt{6}+1=\boldsymbol{7+2\sqrt{6}}\)
(c)
Denominator: \(1.2\times5=6\) and \(10^2\times10^{-8}=10^{-6}\). Denom = \(6\times10^{-6}\).
\(\dfrac{3.6\times10^{-4}}{6\times10^{-6}}=0.6\times10^2=\boldsymbol{6\times10^1}\)
\(\dfrac{3.6\times10^{-4}}{6\times10^{-6}}=0.6\times10^2=\boldsymbol{6\times10^1}\)