Ratio, Proportion, and Scale Drawing: Mathematical Relationships and Scaling
Mathematical reasoning fundamentally relies upon comparing quantitative values through ratios and proportions. A ratio expresses the quantitative comparison between two numbers of the same kind and unit through division, expressed as a : b or a/b. When two ratios are equivalent, they form a proportion (a/b = c/d), embodying the universal fundamental theorem: the product of the extremes equals the product of the means (ad = bc). Applying proportional reasoning enables architects, cartographers, and engineers to create scale drawings where real-world landscapes and structures are faithfully enlarged or reduced. In this foundational lesson, Prep 1 students explore the algebraic properties of ratios, master proportional division in real-life inheritance and commercial partnerships, and calculate scale factors with mathematical rigor.
1. Concept and Properties of Ratio
A ratio is a comparison between two homogeneous mathematical quantities of the same nature and measurement unit by means of division:
Notation: The ratio of quantity a to quantity b is written as a : b or a / b (where b ≠ 0).
The first term a is termed the antecedent, and the second term b is termed the consequent.
| Property | Mathematical Statement | Practical Exemplar |
|---|---|---|
| Invariance under Scaling | Multiplying or dividing both terms of a ratio by a non-zero constant preserves the ratio: (a × c) / (b × c) = a / b. | 12 / 18 = (12 ÷ 6) / (18 ÷ 6) = 2 / 3. |
| Unit Uniformity | Terms must share identical units before simplification. | Ratio of 500 g to 2 kg: 500 g : 2000 g = 1 : 4. |
2. Proportion and Cross-Multiplication
A proportion is an equality between two or more ratios: a / b = c / d.
Fundamental Theorem of Proportion :
In any proportion, the product of extremes equals the product of means:
a × d = b × c.
If 2 / 7 = x / 21 → 7x = 42 → x = 6.
Proportional Division :
Dividing a sum S in the ratio m : n:
Total parts = m + n.
First share = S × [m / (m + n)].
Second share = S × [n / (m + n)].
3. Scale Drawing (Reduction vs. Enlargement)
Scale Drawing Formula: Scale = Drawing Length (DL) / Real Length (RL).
- Reduction (مقياس تصغير): When Scale < 1 (e.g., architectural building blueprints, geographic maps 1 : 100,000).
- Enlargement (مقياس تكبير): When Scale > 1 (e.g., biological microscopic photography of microbes 500 : 1).
Practical Problem-Solving Workshop
Activity 1: Map Scale Problem
The distance between Cairo and Tanta on a map is 9 cm. If the actual distance is 90 km, determine the map scale in simplest form.
Activity 2: Commercial Partnership Division
Two partners invest capital in the ratio 3 : 5. If their annual profit is 64,000 LE, calculate each partner's profit share.
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