Number Work: Whole Numbers, Fractions, Decimals

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Understanding types of numbers and converting between them is fundamental to all mathematics. This topic covers whole numbers, fractions, and decimals — and how to work with each.

Types of Numbers:

Type	Definition	Examples
Natural/Counting numbers	Numbers used for counting	$1, 2, 3, 4, …$
Whole numbers	Natural numbers plus zero	$0, 1, 2, 3, 4, …$
Integers	Whole numbers and their negatives	$-3, -2, -1, 0, 1, 2, 3, …$
Even numbers	Divisible by 2	$2, 4, 6, 8, …$
Odd numbers	Not divisible by 2	$1, 3, 5, 7, …$
Prime numbers	Divisible only by 1 and itself	$2, 3, 5, 7, 11, 13, 17, …$
Composite numbers	Have divisors other than 1 and itself	$4, 6, 8, 9, 10, 12, …$

Note: 1 is neither prime nor composite. 2 is the only even prime number.

Fractions:

A fraction has a numerator (top) and denominator (bottom): $$\frac{\text{numerator}}{\text{denominator}}$$

Type	Example	Meaning
Proper fraction	$3/5$	Numerator < denominator (value < 1)
Improper fraction	$7/4$	Numerator > denominator (value > 1)
Mixed number	$1\frac{3}{4}$	Whole number + proper fraction

Converting Mixed Number to Improper Fraction: $$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4}$$

Converting Improper Fraction to Mixed Number: $$\frac{11}{4} = 2\frac{3}{4}$$

Equivalent Fractions:

Multiply or divide the numerator and denominator by the same number: $$\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$$

To compare fractions, convert to a common denominator: Which is larger: $2/3$ or $3/5$? $2/3 = 10/15$; $3/5 = 9/15$; So $2/3 > 3/5$

Decimals:

Type	Example
Terminating decimal	$0.5, 0.125, 0.75$
Recurring decimal	$0.333… = 0.\overline{3}, 0.1666… = 0.1\overline{6}$

Fraction to Decimal: $3/8 = 3 \div 8 = 0.375$ $1/3 = 0.333…$

Decimal to Fraction: $0.75 = 75/100 = 3/4$ $0.125 = 125/1000 = 1/8$

⚡ Exam Tip (NCEE): To compare fractions quickly, cross-multiply: $2/3 ? 3/5$: $2 \times 5 = 10$ vs $3 \times 3 = 9$. Since $10 > 9$, $2/3 > 3/5$. This is faster than finding a common denominator.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of number operations.

Adding and Subtracting Fractions:

To add/subtract fractions, find a common denominator:

$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

$\frac{3}{5} - \frac{1}{3} = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}$

Multiplying Fractions:

Multiply numerators together and denominators together: $$\frac{2}{3} \times \frac{5}{7} = \frac{2 \times 5}{3 \times 7} = \frac{10}{21}$$

To multiply a fraction by a whole number: $$3 \times \frac{2}{5} = \frac{3 \times 2}{5} = \frac{6}{5} = 1\frac{1}{5}$$

Dividing Fractions:

To divide by a fraction, multiply by its reciprocal: $$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$

Operations with Decimals:

Adding/Subtracting: Align decimal points: $12.5 + 3.47 = 15.97$ $15.6 - 7.23 = 8.37$

Multiplying: Multiply as whole numbers, count decimal places: $1.2 \times 0.3$: $12 \times 3 = 36$; 2 decimal places total → $0.36$

Dividing: Move decimal point to make the divisor a whole number: $4.5 \div 0.3$: Move decimal: $45 \div 3 = 15$

Place Value in Decimals:

$12.345$:

• 1 = Tens (10)
• 2 = Ones (1)
• 3 = Tenths (1/10)
• 4 = Hundredths (1/100)
• 5 = Thousandths (1/1000)

BODMAS/PEMDAS — Order of Operations:

Always perform operations in this order:

1. Brackets / Parentheses
2. Of (powers, roots) / Orders
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Example: $3 + 6 \times (5 + 4) \div 3 - 7$ $= 3 + 6 \times 9 \div 3 - 7$ $= 3 + 54 \div 3 - 7$ $= 3 + 18 - 7$ $= 14$

⚡ Common NCEE Error: When adding fractions like $1/2 + 1/3$, students add numerators AND denominators: $1+1/2+3 = 2/5$ is WRONG. The correct method: $1/2 + 1/3 = (3+2)/6 = 5/6$. Always find a common denominator.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

HCF and LCM:

HCF (Highest Common Factor / GCD): The largest number that divides two or more numbers exactly.

Find HCF of 12 and 18:

• Factors of 12: $1, 2, 3, 4, 6, 12$
• Factors of 18: $1, 2, 3, 6, 9, 18$
• Common factors: $1, 2, 3, 6$
• HCF = 6

Prime Factorisation Method for HCF: $12 = 2 \times 2 \times 3$ $18 = 2 \times 3 \times 3$ HCF = $2 \times 3 = 6$

LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.

Find LCM of 12 and 18: $12 = 2^2 \times 3$ $18 = 2 \times 3^2$ LCM = $2^2 \times 3^2 = 4 \times 9 = 36$

HCF × LCM = Product of the two numbers: $6 \times 36 = 12 \times 18 = 216$ ✓

Uses of HCF and LCM:

• HCF: Finding equal shares, simplifying fractions
• LCM: Finding common time/dates, synchronising cycles

Rounding and Estimation:

Round to the nearest 10: Look at the units digit (≥5 rounds up) $47 \rightarrow 50$; $43 \rightarrow 40$

Round to the nearest 100: Look at the tens digit $347 \rightarrow 300$; $367 \rightarrow 400$

Significant Figures: $4.56$ has 3 significant figures $0.0073$ has 2 significant figures (leading zeros don’t count)

Percentage, Fraction, Decimal Conversions:

Percentage	Fraction	Decimal
1%	1/100	0.01
10%	1/10	0.1
20%	1/5	0.2
25%	1/4	0.25
50%	1/2	0.5
75%	3/4	0.75
100%	1	1.0

Recurring Decimals to Fractions:

$x = 0.\overline{3}$ $10x = 3.\overline{3}$ $10x - x = 3.\overline{3} - 0.\overline{3}$ $9x = 3$ $x = 3/9 = 1/3$

Square Roots and Cube Roots:

$\sqrt{16} = 4$ (since $4 \times 4 = 16$) $\sqrt[3]{27} = 3$ (since $3 \times 3 \times 3 = 27$)

$\sqrt{2} \approx 1.414$ $\sqrt{3} \approx 1.732$ $\sqrt{5} \approx 2.236$

⚡ Extended Tip — Working with Negative Numbers:

• Adding negatives: $-3 + (-5) = -8$ (same sign, add the numbers)
• Subtracting negatives: $-3 - (-5) = -3 + 5 = 2$ (minus a negative = plus)
• Multiplying/dividing negatives: An even number of negatives → positive; an odd number → negative. $(-2) \times (-3) = +6$; $(-2) \times 3 = -6$; $(-2) \times 3 \times (-4) = +24$

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