Topic 4: Ratios and Proportions
🟢 Lite — Quick Review (1h–1d)
Ratios and proportions form an indispensable part of the WASSCE quantitative reasoning syllabus. A ratio is a comparison between two or more quantities of the same kind, expressed in the form a:b or a/b. Crucially, ratios only compare quantities measured in the same units—if a recipe requires 200g of flour and 50g of sugar, the ratio of flour to sugar is 200:50, which simplifies to 4:1. A proportion, on the other hand, is a statement that two ratios are equal—for example, 2:3 = 6:9. When we say two quantities are “directly proportional,” we mean that as one increases, the other increases at a constant rate, maintaining the same ratio.
Understanding the relationship between ratios and fractions is fundamental. The ratio a:b is equivalent to the fraction a/b. When asked to share an amount in a given ratio, you first find one part by dividing the total by the sum of ratio parts, then multiply by each individual ratio number. These concepts appear extensively in problems involving sharing money, dividing inheritance, mixing solutions, and interpreting map scales.
Key Facts:
- Ratio a:b means “a parts to b parts” of the same whole
- Total parts in ratio a:b = a + b
- If x:y = 2:3 and x = 10, then y = 15 (because 2/3 = 10/15)
- Direct proportion: as one quantity increases, the other increases at constant rate (a/b = c/d)
- Inverse proportion: as one quantity increases, the other decreases (a × b = constant)
- Scale factor: the multiplier used to convert between actual size and scaled size
- Map scale: given as ratio (e.g., 1:50,000 means 1 unit on map = 50,000 same units in reality)
⚡ Exam Tip: In WASSCE questions involving “sharing money in a ratio,” always find the value of ONE ratio part first by dividing the total by the sum of ratio parts.
🟡 Standard — Regular Study (2d–2mo)
Sharing Quantities in a Given Ratio
When GH₵ 2,400 is shared between Ama and Kofi in the ratio 3:5:
Step 1: Total ratio parts = 3 + 5 = 8 Step 2: Value of 1 part = GH₵ 2,400 ÷ 8 = GH₵ 300 Step 3: Ama receives 3 × 300 = GH₵ 900 Step 4: Kofi receives 5 × 300 = GH₵ 1,500 Verification: 900 + 1,500 = 2,400 ✓
Dividing a Quantity in Compound Ratios
When dividing among three or more people in a compound ratio:
GH₵ 5,600 is divided among Abena, Ama, and Kofi in the ratio 2:3:5:
Total parts = 2 + 3 + 5 = 10 Value of 1 part = 5,600 ÷ 10 = 560
- Abena: 2 × 560 = GH₵ 1,120
- Ama: 3 × 560 = GH₵ 1,680
- Kofi: 5 × 560 = GH₵ 2,800
Direct Proportion
Two quantities are directly proportional if they increase or decrease in the same ratio. If 8 textbooks cost GH₵ 640, find the cost of 5 textbooks:
Method 1 — Unitary method: Cost of 1 textbook = 640 ÷ 8 = GH₵ 80 Cost of 5 textbooks = 5 × 80 = GH₵ 400
Method 2 — Ratio method: 8:5 = 640:x → 8/5 = 640/x → x = (640 × 5) ÷ 8 = GH₵ 400
Inverse Proportion
Two quantities are inversely proportional if one increases as the other decreases. If 12 workers can complete a job in 20 days, how long will 15 workers take?
Since workers increase, days decrease (inverse): 12 workers × 20 days = 15 workers × x days 240 = 15x → x = 240 ÷ 15 = 16 days
Comparison Table: Direct vs Inverse Proportion
| Property | Direct Proportion | Inverse Proportion |
|---|---|---|
| Relationship | a ∝ b | a ∝ 1/b |
| Equation | a = kb (k = constant) | a × b = k |
| Graph | Straight line through origin | Rectangular hyperbola |
| Example | Cost ∝ number of items | Workers ∝ 1/time for job |
Map Scale Problems
A map is drawn with scale 1:50,000. If two towns are 8 cm apart on the map:
- Actual distance = 8 × 50,000 = 400,000 cm = 4 km
If two towns are 6 km apart in reality, what is their distance on the map?
- 6 km = 600,000 cm
- Map distance = 600,000 ÷ 50,000 = 12 cm
Common Mistakes to Avoid:
- Mixing units before forming ratios (convert all to same unit first)
- Forgetting to add ratio parts when sharing quantities
- Confusing direct with inverse proportion
- In scale problems, forgetting that scale is a ratio of lengths, not areas
- For compound ratios, dividing the total by the wrong number of parts
Problem-Solving Strategy:
- Determine whether the problem involves sharing, direct proportion, or inverse proportion
- For sharing: find one part, then multiply
- For direct proportion: set up equation a₁/a₂ = b₁/b₂ or use unitary method
- For inverse proportion: set up equation a₁ × b₁ = a₂ × b₂
- Always check that your answer is sensible within the problem context
🔴 Extended — Deep Study (3mo+)
Historical Context: The Golden Ratio
The ratio φ (phi) = (1 + √5)/2 ≈ 1.618:1, known as the Golden Ratio, has fascinated mathematicians since ancient Greece. Euclid defined it as the division of a line into “extreme and mean ratio”—the point that divides a segment such that the ratio of the whole segment to the longer part equals the ratio of the longer part to the shorter part. This ratio appears throughout nature (spiral shells, flower petals,人体 proportions) and classical architecture, including the Parthenon. Understanding ratios therefore connects modern WASSCE problems to millennia of mathematical tradition.
Proportional Parts and Internal Division
When a line segment is divided internally in a given ratio:
Point P divides AB in the ratio m:n. If A is at (x₁, y₁) and B is at (x₂, y₂), coordinates of P are: P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
For example, P divides AB in ratio 2:3 where A(2,3) and B(8,9): P = ((2×8 + 3×2)/(2+3), (2×9 + 3×3)/(2+3)) P = ((16+6)/5, (18+9)/5) = (22/5, 27/5) = (4.4, 5.4)
Rates and Unit Rates
A rate is a ratio comparing two quantities of different kinds (e.g., cedis per kilogram, kilometres per hour). A unit rate has a denominator of 1.
If a car travels 360 km in 4 hours, its average speed is: 360 km ÷ 4 hours = 90 km/h
Converting between rates: to convert 72 km/h to m/s: 72 km/h × (1000 m/km) ÷ (3600 s/h) = 72,000 ÷ 3,600 = 20 m/s
Compound Ratio Problems
“A mixture contains wine and water in the ratio 5:3. When 10 litres of water are added, the ratio becomes 5:4. Find the original quantities.”
Let original wine = 5x litres and water = 3x litres. After adding 10L water: (5x)/(3x+10) = 5/4 Cross-multiplying: 20x = 15x + 50 → 5x = 50 → x = 10 Original: Wine = 50L, Water = 30L
Proving Direct and Inverse Variation
To prove y varies directly as x: show that y/x = constant To prove y varies inversely as x: show that xy = constant
If y = kx (direct), or y = k/x (inverse), where k is the constant of proportionality.
Proportion in Similar Figures
For similar triangles and geometrically similar shapes:
- Corresponding sides are in the same ratio (scale factor k)
- Areas are in ratio k²
- Volumes are in ratio k³
If two similar cylinders have heights in ratio 3:5:
- Their radii are also in ratio 3:5
- Their surface areas are in ratio 9:25
- Their volumes are in ratio 27:125
WASSCE Examination Patterns:
The WASSCE quantitative reasoning paper commonly features:
- Sharing money or goods in given ratios (Objective)
- Rate problems involving speed, time, and distance (Theory)
- Map scale calculations (Objective and Theory)
- Mixture problems involving ratios (Theory)
- Problems involving direct and inverse variation (Theory)
⚡ Pro Exam Tip: In WASSCE, always check whether a ratio has been simplified before using it in calculations. An unsimplified ratio like 8:12 might actually be 2:3. Also, in “sharing” problems, verify your answer by adding all shares—they must equal the total amount.
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