Fractions, Decimals, and Percentages
Fractions, decimals, and percentages are three different ways of representing the same concept — a part of a whole. For the Post-UTME examination, mastery of conversions between these forms, operations on each type, and their practical applications is absolutely essential. These questions appear in nearly every Post-UTME paper and are a common source of careless errors.
Fractions: Definitions and Types
A fraction represents a part of a whole, expressed as a/b where:
- a = numerator (the number of parts we have)
- b = denominator (the total number of equal parts) — b ≠ 0
Types of Fractions
| Type | Description | Example |
|---|---|---|
| Proper Fraction | Numerator < denominator | 3/5, 7/8 |
| Improper Fraction | Numerator ≥ denominator | 9/5, 13/4 |
| Mixed Number | Integer + proper fraction | 3½, 2¾ |
| Equivalent Fractions | Same value, different form | 2/4 = 1/2 |
| Unit Fraction | Numerator = 1 | 1/3, 1/7 |
Converting Between Forms
Improper Fraction → Mixed Number: Divide numerator by denominator; quotient is the integer part, remainder becomes the new numerator.
- Example: 17/5 = 17 ÷ 5 = 3 remainder 2 → 3⅖
Mixed Number → Improper Fraction: (Integer × denominator) + numerator = new numerator; denominator stays the same.
- Example: 3⅖ = (3 × 5) + 2 = 17 → 17/5
Operations on Fractions
Addition and Subtraction
Rule: Fractions must have a common denominator before adding or subtracting.
Example: 3/4 + 2/5
- LCD of 4 and 5 = 20
- 3/4 = 15/20; 2/5 = 8/20
- 15/20 + 8/20 = 23/20 = 1⅜
Multiplication
Multiply numerators together and denominators together. Simplify.
- Example: 3/5 × 7/8 = (3 × 7)/(5 × 8) = 21/40
Division
Rule: Multiply by the reciprocal (flip the second fraction).
- Example: 3/4 ÷ 5/7 = 3/4 × 7/5 = 21/20 = 1 1/20
Decimals
Types of Decimals
- Terminating Decimals: End after a finite number of digits (e.g., 0.5, 0.125)
- Repeating (Recurring) Decimals: One or more digits repeat forever (e.g., 0.333… = 0.3)
- Non-terminating, Non-repeating: Irrational numbers (e.g., √2, π)
Converting Fractions to Decimals
Simply divide the numerator by the denominator.
- 1/8 = 1 ÷ 8 = 0.125 (terminating)
- 1/3 = 1 ÷ 3 = 0.333… = 0.3 (repeating)
Converting Decimals to Fractions
- Terminating: Count decimal places; put over 10ⁿ; simplify.
- Example: 0.375 = 375/1000 = 3/8 (divide by 125)
- Repeating: Use algebra. Let x = 0.3… → 10x = 3.3… → 9x = 3 → x = 1/3
Percentages
Percent means “per hundred” — a fraction with denominator 100.
Conversions
- Fraction → Percentage: Multiply by 100 and add %
- Example: 3/5 = (3/5) × 100 = 60%
- Decimal → Percentage: Multiply by 100
- Example: 0.875 = 0.875 × 100 = 87.5%
- Percentage → Fraction: Divide by 100 and simplify
- Example: 75% = 75/100 = 3/4
Percentage Calculations
Finding a percentage of a number:
- Method: (Percentage/100) × Number
- Example: 35% of 240 = (35/100) × 240 = 0.35 × 240 = 84
Finding what percentage one number is of another:
- Method: (Part/Whole) × 100
- Example: What % is 18 of 60? = (18/60) × 100 = 30%
Percentage Increase and Decrease
Increase: New = Original × (1 + r) where r = percentage increase/100 Decrease: New = Original × (1 − r) where r = percentage decrease/100
Example (Increase): A price of ₦500 increases by 15%. New price = 500 × 1.15 = ₦575 Example (Decrease): A population of 2,000 decreases by 8%. New population = 2000 × 0.92 = 1,840
Successive Percentage Changes
Example: A value increases by 20%, then decreases by 10%:
- After 20% increase: 100 × 1.20 = 120
- After 10% decrease: 120 × 0.90 = 108 (net +8%)
Key Post-UTME Exam Facts
- Percentage to fraction: Divide by 100 and simplify
- Percentage change: Don’t add — multiply successive factors: New = Old × (1+r₁)(1+r₂)
- Successive changes: A 10% increase followed by 10% decrease does NOT return to original (results in 0.99 × original)
- Profit % = (Profit/Cost Price) × 100; Loss % = (Loss/Cost Price) × 100
- Discount: Selling Price = Marked Price − Discount Amount
- ⚡ Exam tip: When a fraction won’t convert to a clean decimal, expect a recurring decimal. Learn 1/3=0.3, 2/3=0.6, 1/6=0.16, 5/6=0.83, 1/7=0.142857, 1/9=0.1, 1/11=0.09
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
- Fractions: LCD before adding/subtracting; multiply numerators and denominators for multiplication; multiply by reciprocal for division
- Decimals: 0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4, 0.125 = 1/8 — memorize these
- Converting: Fraction → Decimal (divide); Decimal → Fraction (count decimal places)
- Percentage calculations: (Part/Whole) × 100 for percentage; (r/100) × N for finding percentage of N
- Percentage change: Multiply successive factors — don’t just add/subtract percentages
- ⚡ Exam tip: Always check if your answer is reasonable — an increase should give a larger number
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Common Fraction-Decimal-Percentage Equivalents
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 2/3 | 0.666… | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1666… | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 1/16 | 0.0625 | 6.25% |
| 1/20 | 0.05 | 5% |
Applications: Discount, Profit, and Loss
Profit and Loss
- Cost Price (CP): The price at which an item is bought
- Selling Price (SP): The price at which an item is sold
- Profit = SP − CP; Loss = CP − SP
- Profit % = (Profit/CP) × 100; Loss % = (Loss/CP) × 100
Example: A trader buys 50 bags at ₦3,000 each and sells at ₦3,600 each.
- Total CP = ₦150,000; Total SP = ₦180,000
- Profit = ₦30,000; Profit % = (30,000/150,000) × 100 = 20%
Discount
- Marked Price (MP): The original listed price
- Selling Price after Discount = MP × (1 − Discount%)
Example: MP = ₦5,000, Discount = 15% → SP = 5000 × 0.85 = ₦4,250
🔴 Extended — Deep Study (3mo+)
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Recurring Decimals and Their Fraction Equivalents
Method: Algebra
Convert 0.63 (where 6 and 3 repeat: 0.636363…) to a fraction. Let x = 0.6363… Multiply by 100: 100x = 63.6363… Subtract x: 99x = 63 → x = 63/99 = 7/11
Direct Formulas
- 0.AAAA… (1 digit repeats) = A/9
- 0.ABABAB… (2 digits repeat) = AB/99
- 0.1̅6 = (16−1)/90 = 15/90 = 1/6
Compound Interest
Formula: A = P(1 + r/n)^(nt) Where: A = Amount, P = Principal, r = annual rate (decimal), n = times compounded/year, t = years
Example: ₦10,000 at 10% per annum compound interest for 3 years: A = 10,000(1.10)³ = 10,000 × 1.331 = ₦13,310 (Compare Simple Interest: SI = P × r × t / 100 = ₦3,000; Total = ₦13,000)
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