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If 5 workers build a wall in 12 days, will 10 workers take 24 days or 6 days? Proportional reasoning distinguishes between quantities that increase to...
If 5 workers build a wall in 12 days, will 10 workers take 24 days or 6 days? Proportional reasoning distinguishes between quantities that increase together (Direct Proportion) and quantities that balance each other out (Inverse Proportion).
Why This Chapter Matters
In Class 8 Mathematics, "Proportional Reasoning" delivers rigorous mathematical reasoning, algebraic precision, and geometric mastery aligned with the 2026–27 NCERT curriculum.
Before You Begin (Prerequisites)
- Basic ratios and equivalent fractions.
- Unitary method.
- Speed, distance, and time relationships.
What You Will Learn (Core Objectives)
- Define Direct Proportion ($x \propto y \implies \frac{x}{y} = k$).
- Define Inverse Proportion ($x \propto \frac{1}{y} \implies x \times y = k$).
- Identify whether real-life scenarios are direct or inverse proportions.
- Solve multi-step problems involving maps, fuel consumption, and scale models.
- Calculate work-and-time problems using inverse proportional reasoning.
Chapter Roadmap & Progression
1
1. Direct Proportion (Constant Rati...
2
2. Inverse Proportion (Constant Pro...
3
3. Map Scales and Scale Drawings
Complete Concept Guide (100% Curriculum Coverage)
1. Direct Proportion (Constant Ratio)
Two quantities $x$ and $y$ are in Direct Proportion if an increase in $x$ causes a proportional increase in $y$, such that their ratio remains constant: $\mathbf{\frac{x}{y} = k}$ (constant). Example: Number of articles purchased and total cost; distance travelled and petrol consumed.
2. Inverse Proportion (Constant Product)
Two quantities $x$ and $y$ are in Inverse Proportion if an increase in $x$ causes a proportional decrease in $y$, such that their product remains constant: $\mathbf{x \times y = k}$ (constant). Example: Number of workers and time taken to finish a job; vehicle speed and travel time.
3. Map Scales and Scale Drawings
Architectural blueprints and world maps rely on direct proportions: a scale of $1\text{ cm} : 50\text{ km}$ means every centimeter measured on paper represents $50\text{ km}$ in reality.
Visual Learning & Conceptual Map
Proportional Reasoning Mathematical Matrix
Core formulas, geometric proofs, and computational algorithms
Mathematical Framework
1. Direct Proportion (Constant Ratio) • 2. Inverse Proportion (Constant Product) • 3. Map Scales and Scale Drawings
Chapter Summary & 10 Key Takeaways
Takeaway 1
Direct Proportion: $\frac{x}{y} = k$ (Both quantities increase or decrease together).
Takeaway 2
Inverse Proportion: $x \times y = k$ (One increases as the other decreases proportionally).
Takeaway 3
Speed and Time: Inversely proportional for a fixed distance.
Takeaway 4
Workers and Time: Inversely proportional for a fixed task.
Takeaway 5
Unitary Method: Finding the value of one unit first to solve proportions.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
If 14 meters of cloth costs ₹1890, what is the cost of 6 meters of the same cloth?
Reveal Answer & Explanation
Answer: Direct proportion: $\frac{1890}{14} = \frac{x}{6} \implies x = \frac{1890 \times 6}{14} = 135 \times 6 = ₹ 810$.
Direct ratio.
2
If 15 workers can build a wall in 48 hours, how many workers are needed to do the same work in 30 hours?
Reveal Answer & Explanation
Answer: Inverse proportion: $15 \times 48 = x \times 30 \implies x = \frac{720}{30} = 24\text{ workers}$.
Constant product.
3
A car travels 432 km on 36 liters of petrol. How far will it travel on 25 liters?
Reveal Answer & Explanation
Answer: Direct proportion: $\frac{432}{36} = 12\text{ km/L}$. Distance on 25 L $= 12 \times 25 = 300\text{ km}$.
Km per liter times 25.
4
State whether the speed of a vehicle and the time taken to cover a fixed distance are in direct or inverse proportion.
Reveal Answer & Explanation
Answer: Inverse proportion, because as speed increases, the travel time decreases proportionally.
Faster speed means less time.
5
On a map, 1 cm represents 8 km. If two towns are 72 km apart, what is the distance between them on the map?
Reveal Answer & Explanation
Answer: Direct proportion: $\frac{72}{8} = 9\text{ cm}$.
Divide 72 by 8.
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