Chapter 10: Proportional Reasoning – 2

1. Proportionality Recap

In direct proportion, corresponding quantities maintain a constant ratio. If one quantity is multiplied by 3, the other is also multiplied by 3.

2. Ratios in Maps

A map scale connects map distance with actual distance. For example, a scale of 1 cm:5 km means 7 cm represents 35 km. Units must be kept consistent.

3. Ratios with More Than Two Terms

A ratio such as 2:3:5 compares three quantities. If the common multiplier is 4, the corresponding quantities are 8:12:20.

4. Dividing a Whole in a Given Ratio

To divide ₹500 in the ratio 2:3, total parts=5, one part=₹100, so the shares are ₹200 and ₹300.

5. A Slice of the Pie

A pie chart represents the whole by 360°. If a category is 25% of the whole, its sector angle is 25/100×360°=90°.

6. Direct and Inverse Proportion

In direct proportion, quantities change together by the same scale factor. In inverse proportion, one quantity increases while the other decreases so that their product remains constant under the given conditions. For a fixed job, if 4 workers take 12 days, the ideal inverse-proportion model gives 8 workers 6 days.

🧠 MCQs

1. A 1:5 map scale means 3 cm represents: A) 3 km B) 8 km C) 15 km D) 20 km
2. ₹600 divided in ratio 2:4 gives: A) ₹100,₹500 B) ₹200,₹400 C) ₹300,₹300 D) ₹150,₹450
3. A full pie chart represents: A) 90° B) 180° C) 270° D) 360°
4. If x and y are inversely proportional, xy is: A) constant B) zero C) always 1 D) increasing

Answers: 1-C, 2-B, 3-D, 4-A.

📌 Case Study

A map uses 1 cm to represent 4 km. Two towns are 8.5 cm apart on the map. Find their actual distance. Then explain how the scale is a proportional relationship.

🔥 HOTS

1. Divide ₹840 among three people in the ratio 2:3:7.
2. Create a pie chart angle for a category representing 35%.
3. Explain why “more workers means fewer days” is not automatically inverse proportion unless the work and productivity assumptions remain fixed.

✍️ Practice

1. Divide 450 in ratio 4:5.
2. Convert 30% into a pie-chart angle.
3. Explain direct and inverse proportion with examples.
4. Solve a map-scale problem with consistent units.

🏆 Mastery Checklist

• ☐ I can use scale ratios.
• ☐ I can divide quantities in a given ratio.
• ☐ I can interpret pie-chart angles.
• ☐ I can distinguish direct from inverse proportion.

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