Fractions, Decimals and Percentages

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Fractions, decimals, and percentages are different ways of expressing the same concept: a part of a whole.

Fractions: A fraction has a numerator (top number) and denominator (bottom number). $$\frac{a}{b} = a \div b$$

Types of Fractions:

• Proper fraction: Numerator < denominator (e.g., $\frac{3}{4}$, $\frac{5}{8}$). Value < 1.
• Improper fraction: Numerator > denominator (e.g., $\frac{7}{4}$, $\frac{11}{3}$). Value > 1.
• Mixed number: Whole number + proper fraction (e.g., $2\frac{1}{3}$, $1\frac{3}{4}$)

Converting between forms:

• Improper to mixed: $\frac{11}{4} = 2\frac{3}{4}$ (11 ÷ 4 = 2 remainder 3)
• Mixed to improper: $2\frac{3}{4} = \frac{11}{4}$ (2 × 4 + 3 = 11)

Decimals: Decimals are fractions with denominators of 10, 100, 1000, etc. $$\frac{3}{10} = 0.3, \quad \frac{47}{100} = 0.47, \quad \frac{123}{1000} = 0.123$$

Types of Decimals:

• Terminating: Ends after finite digits (e.g., 0.5, 0.125)
• Recurring: Repeats forever (e.g., 0.333… = $0.\overline{3}$, $0.1666… = 0.1\overline{6}$)
• Non-terminating, non-recurring: Irrational numbers (e.g., $\pi = 3.14159…$, $\sqrt{2} = 1.41421…$)

Percentages: “Per cent” means “per hundred”: $$25% = \frac{25}{100} = 0.25 = \frac{1}{4}$$

⚡ WAEC Tip: Always check whether your answer should be in fraction, decimal, or percentage form. WAEC questions often require converting between forms. If a fraction doesn’t simplify nicely (like $\frac{1}{3}$), the decimal will be recurring.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Operations with Fractions:

Addition and Subtraction: $$\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} \quad \text{(same denominator)}$$ $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \quad \text{(different denominators)}$$

Example: $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$

Multiplication: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$

Example: $\frac{3}{5} \times \frac{2}{7} = \frac{6}{35}$

Division: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

Example: $\frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} = 2\frac{1}{10}$

Converting Recurring Decimals to Fractions:

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

For mixed recurring: $x = 0.1\overline{6}$ $10x = 1.\overline{6}$ $10x - x = 1.\overline{6} - 0.\overline{6}$ — WAIT: $x = 0.1\overline{6}$, so $10x = 1.\overline{6}$, and $x = 0.1\overline{6}$ Actually: $10x = 1.6\overline{6}$, $100x = 16.\overline{6}$ $100x - 10x = 16.\overline{6} - 1.6\overline{6} = 15$ $90x = 15$, $x = \frac{15}{90} = \frac{1}{6}$

Percentage Calculations:

Problem	Method
Find X% of Y	$Y \times \frac{X}{100}$
Express a as % of b	$\frac{a}{b} \times 100$
Increase by X%	Multiply by $1 + \frac{X}{100}$
Decrease by X%	Multiply by $1 - \frac{X}{100}$

Compound Interest Formula: $$A = P\left(1 + \frac{r}{100}\right)^n$$ Where: A = amount, P = principal, r = rate per period, n = number of periods

Simple Interest: $$I = \frac{PRT}{100}$$ Where: I = interest, P = principal, R = rate, T = time (years)

⚡ Common Student Mistakes: Forgetting to convert denominators before adding/subtracting fractions. Confusing “of” with multiplication when calculating percentages (e.g., “20% of 50” = 10, not 10%). Forgetting that percentage increase/decrease is calculated on the ORIGINAL amount.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Rational and Irrational Numbers:

Rational numbers: Can be expressed as $\frac{p}{q}$ where p and q are integers, q ≠ 0. This includes:

• All integers (e.g., 5 = $\frac{5}{1}$)
• All fractions (proper and improper)
• Terminating decimals (e.g., 0.125 = $\frac{125}{1000}$)
• Recurring decimals (e.g., $0.\overline{3} = \frac{1}{3}$)

Irrational numbers: Cannot be expressed as $\frac{p}{q}$. They have non-terminating, non-recurring decimal expansions.

• $\pi = 3.1415926535…$
• $\sqrt{2} = 1.4142135623…$
• $e = 2.7182818284…$
• Golden ratio $\phi = \frac{1+\sqrt{5}}{2} = 1.618…$

The Real Number System: $$\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}’$$ (Rationals ∪ Irrationals)

Operations on Decimals:

Addition: $$12.35 + 7.4 = 12.35 + 07.40 = 19.75$$

Multiplication: $$3.2 \times 1.5 = \frac{32}{10} \times \frac{15}{10} = \frac{480}{100} = 4.80 = 4.8$$

Dividing decimals: To divide 7.5 by 0.25: $7.5 \div 0.25 = \frac{7.5}{0.25} = \frac{750}{25} = 30$

Standard Form (Scientific Notation): Very large or very small numbers are written as: $$a \times 10^n \quad \text{where } 1 \leq a < 10 \text{ and } n \in \mathbb{Z}$$

Examples:

• $5,200,000 = 5.2 \times 10^6$
• $0.000042 = 4.2 \times 10^{-5}$
• $3.7 \times 10^3 = 3700$
• $2.1 \times 10^{-2} = 0.021$

Calculations with Standard Form: $$(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$$ $$\frac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$$

Approximation and Significant Figures:

Decimal Places	Round to…	Example
1 d.p.	Hundredths	3.46 → 3.5
2 d.p.	Thousandths	7.834 → 7.83
3 d.p.	Ten-thousandths	2.4567 → 2.457

Significant Figures:

1. First non-zero digit is the first significant figure
2. Count all digits from first to last non-zero
3. Trailing zeros in decimal numbers ARE significant

Examples:

• 3.50 has 3 significant figures
• 0.006 has 1 significant figure
• 4500 has 2 significant figures (could be written as 4.5 × 10³)

Percentage Error: $$\text{Percentage Error} = \left|\frac{\text{Approximate} - \text{Exact}}{\text{Exact}}\right| \times 100%$$

Order of Operations (BODMAS):

1. Brackets (parentheses)
2. Of (multiplication in context)
3. Division
4. Multiplication
5. Addition
6. Subtraction

Example: $3 + 4 \times 2 = 3 + 8 = 11$ (NOT $3 + 4 = 7 \times 2 = 14$)

Fractions in Real-World WAEC Problems:

Problem 1: A trader buys goods for ₦12,000 and sells at a profit of 25%. What is the selling price? $$SP = 12000 \times \left(1 + \frac{25}{100}\right) = 12000 \times 1.25 = \text{₦15,000}$$

Problem 2: A car depreciates by 15% per year. If its initial value is ₦3,000,000, what is its value after 2 years? $$V = 3000000 \times \left(1 - \frac{15}{100}\right)^2 = 3000000 \times (0.85)^2 = 3000000 \times 0.7225 = \text{₦2,167,500}$$

Problem 3: Express $0.3\overline{6}$ as a fraction. Let $x = 0.3\overline{6}$ $10x = 3.6\overline{6}$, $100x = 36.\overline{6}$ $100x - 10x = 36.\overline{6} - 3.6\overline{6} = 33$ $90x = 33$ $x = \frac{33}{90} = \frac{11}{30}$

Problem 4: Simplify $\frac{2\frac{1}{3} \times 1\frac{1}{2}}{1\frac{1}{4}}$ Convert to improper fractions: $\frac{7}{3} \times \frac{3}{2} \div \frac{5}{4} = \frac{21}{6} \times \frac{4}{5} = \frac{84}{30} = \frac{14}{5} = 2\frac{4}{5}$

⚡ WAEC Examination Patterns: Simplify complex fractions. Convert between all three forms (fraction, decimal, percentage). Calculate percentage increase/decrease and profit/loss. Express answers in standard form. Solve problems involving simple and compound interest. Convert recurring decimals to fractions.