Topic 3: Fractions, Decimals and Percentages

🟢 Lite — Quick Review (1h–1d)

Fractions, decimals, and percentages represent three faces of the same mathematical concept: the division of whole quantities into parts. A fraction such as ¾ expresses a part-to-whole relationship where 3 parts are taken from a total of 4 equal parts. A decimal like 0.75 is simply another way of writing this same quantity, using base-10 place values. A percentage—75%—means “per hundred,” so 75% is literally 75 hundredths, equivalent to the fraction 75/100 or, in simplest form, 3/4. Mastery of converting between these three forms is absolutely essential for the WASSCE quantitative reasoning paper.

The conversion relationships are straightforward. To change a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. To change a percentage to a fraction, write it over 100 and simplify. To change a fraction to a percentage, either convert to decimal first then multiply by 100, or multiply numerator and denominator so the denominator becomes 100. These operations appear constantly in examination questions, often embedded within more complex problems involving ratios, profit and loss, or statistical averages.

Key Facts:

• Fraction to decimal: divide numerator ÷ denominator
• Decimal to percentage: multiply by 100 (e.g., 0.375 × 100 = 37.5%)
• Percentage to fraction: write over 100, simplify (e.g., 75% = 75/100 = 3/4)
• Common equivalent values to memorise: ½ = 0.5 = 50%; ¼ = 0.25 = 25%; ¾ = 0.75 = 75%; 1/3 ≈ 0.333 = 33.3%; 2/3 ≈ 0.667 = 66.7%; 1/8 = 0.125 = 12.5%
• To add or subtract fractions: find a common denominator
• To multiply fractions: multiply numerators together and denominators together
• To divide fractions: multiply by the reciprocal of the divisor

⚡ Exam Tip: When a WASSCE question involves money (cedis and pesewas), work in pesewas exclusively. Never mix cedis and pesewas in calculations—convert everything to the smaller unit first.

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🟡 Standard — Regular Study (2d–2mo)

Operations with Fractions

Addition and subtraction require a common denominator. There are two approaches:

Method 1 — LCM of denominators: Add 2/3 and 5/8:

• LCM of 3 and 8 = 24
• Convert: 2/3 = 16/24, 5/8 = 15/24
• Add: 16/24 + 15/24 = 31/24 = 1 7/24

Method 2 — Cross-multiplication for two fractions: a/b + c/d = (ad + bc) / bd

For multiplication, simply multiply across: (2/3) × (5/8) = (2×5)/(3×8) = 10/24 = 5/12

For division, invert and multiply: (2/3) ÷ (5/8) = (2/3) × (8/5) = 16/15 = 1 1/15

Comparison Table: Fraction Operations

Operation	Rule	Example
Addition	Find common denominator, then add numerators	1/4 + 1/6 = 3/12 + 2/12 = 5/12
Subtraction	Find common denominator, then subtract numerators	2/3 - 1/4 = 8/12 - 3/12 = 5/12
Multiplication	Multiply all numerators, multiply all denominators	3/5 × 2/7 = 6/35
Division	Multiply by reciprocal of divisor	3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2.1

Percentage Change

Percentage increase and decrease are computed using the formula:

• Percentage increase = (actual increase ÷ original value) × 100
• Percentage decrease = (actual decrease ÷ original value) × 100

If a shirt costing GH₵ 80 is sold for GH₵ 100, the percentage profit is:

• Profit = GH₵ 100 - GH₵ 80 = GH₵ 20
• Percentage profit = (20 ÷ 80) × 100 = 25%

To reverse a percentage change (find the original value before a percentage increase or decrease):

• If a price increases by 20% to GH₵ 120, the original was 120 ÷ 1.20 = GH₵ 100
• If a price decreases by 15% to GH₵ 85, the original was 85 ÷ 0.85 = GH₵ 100

Compound Percentages

When multiple percentage changes occur sequentially, do NOT simply add or subtract percentages. Instead, use multipliers:

A value of GH₵ 200 first increases by 10%, then decreases by 10%:

• After increase: 200 × 1.10 = GH₵ 220
• After decrease: 220 × 0.90 = GH₵ 198

Notice this is NOT equal to the original GH₵ 200—this is a common examination trap.

Common Mistakes to Avoid:

1. Confusing “percentage of” with “percentage increase”
2. Adding percentages that apply to different base values
3. Failing to simplify fractions to lowest terms
4. Misplacing decimal points when converting between forms
5. Forgetting that 100% = 1 (the whole)

Problem-Solving Strategy:

1. Identify all given values and what is being asked
2. Convert all quantities to the same form (fraction, decimal, or percentage)
3. Perform the required operation
4. Convert back to the requested form if different from step 2
5. Always check whether your answer is sensible (e.g., a percentage cannot exceed 100% unless multiple quantities are being compared)

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🔴 Extended — Deep Study (3mo+)

Historical Context: The Origins of Percentage

The concept of percentage—derived from the Latin “per centum” meaning “by the hundred”—emerged from the need to assess taxes, interest, and trade profits in ancient Rome. Roman emperors levied taxes expressed as fractions of 100, and by the medieval period, percentage notation had become standard in commercial transactions across Europe. The decimal system, which we now use to express percentages, was formalised by the Flemish mathematician Simon Stevin in 1585, revolutionising financial calculations.

Recursive Percentage Problems

A particularly challenging WASSCE question type involves finding original values after multiple percentage changes:

“A number is increased by 30% to 169. Find the original number.” Solution: Let original = x. x × 1.30 = 169, so x = 169 ÷ 1.30 = 130

More complex: “The price of a commodity first increased by 20%, then decreased by 15% to GH₵ 204. Find the original price.”

• Let original = P
• After 20% increase: P × 1.20
• After 15% decrease: P × 1.20 × 0.85 = P × 1.02
• 1.02P = 204 → P = 204 ÷ 1.02 = GH₵ 200

Percentage Points vs Percentage

In statistical and financial contexts, distinguishing between “percentage points” and “percentages” is crucial:

• If interest rates rise from 10% to 12%, the increase is 2 percentage points
• The percentage increase is (2 ÷ 10) × 100 = 20%

This distinction appears in WASSCE questions involving bank rates, inflation, and statistical comparisons.

Recurring Decimals

Some fractions convert to recurring decimals—decimals where digits repeat indefinitely:

• 1/3 = 0.333… = 0.3̅
• 2/7 = 0.285714285714… = 0.285714̅
• 1/11 = 0.090909… = 0.09̅

To convert a recurring decimal to a fraction, use the subtraction method:

• Let x = 0.3̅ = 0.333…
• Multiply by 10: 10x = 3.333…
• Subtract: 10x - x = 3.333… - 0.333… → 9x = 3 → x = 3/9 = 1/3

Advanced Worked Example:

“Three traders shared a profit of GH₵ 4,800 in the ratio 3 : 5 : 8. What fraction of the profit did the first trader receive?”

Total ratio parts = 3 + 5 + 8 = 16 First trader’s share = 3/16 of GH₵ 4,800 = (3/16) × 4800 = GH₵ 900

This problem combines ratio and fraction concepts, a common WASSCE examination format.

WASSCE Examination Patterns:

The WASSCE quantitative reasoning paper frequently tests:

1. Conversion between fractions, decimals, and percentages (Objective paper)
2. Percentage profit and loss calculations (Theory paper)
3. Successive percentage changes (Objective and Theory)
4. Ratio and proportion problems involving fractions
5. Word problems requiring identification of the base quantity

⚡ Pro Exam Tip: In the WASSCE, if a question asks “express as a fraction in its simplest form,” always reduce to lowest terms by dividing numerator and denominator by their GCD. For percentage questions, identify the original/base value first—it is always the value BEFORE the change occurred.

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