Points to Remember: Proportional Reasoning-1 | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

Introduction

• In daily life, digital images are often resized or rotated.

• Some resized images look similar (same shape, different size), while others look distorted (shape changes).

• The difference depends on whether the width and height change by the same factor or not.

Observing Similarity in Change

• Similar images → Width and height change by the same multiplication factor.

• Distorted images → Width and height change by different factors.

Examples:

1. Image A vs. C: Both width & height reduced by ½ → Similar.

2. Image A vs. B: Width factor = 2/3, Height factor = ½ → Distorted.

3. Image A vs. D: Both width & height increased by 3/2 → Similar.

4. Image A vs. E: Width factor = 1, Height factor = 3/2 → Distorted.

Conclusion:

• Proportional Change (same factor) → Shape remains same.

• Non-proportional Change (different factors) → Shape looks distorted.

Ratios

• Definition: A ratio compares two quantities of the same kind.

• Form: a : b → “a is to b”.

• Example: Width : Height of Image A = 60 : 40.

Simplifying Ratios

• Divide both terms by their HCF.

• Example:

  • 60 : 40 → HCF = 20 → 3 : 2.

  • 90 : 60 → HCF = 30 → 3 : 2.

Ratios in simplest form make comparisons easier.

Proportional Ratios

• Two ratios are proportional if they simplify to the same simplest form.

• Written as:

  • a : b :: c : d (read as “a is to b as c is to d”).

• Rule: If a : b = c : d, then ad = bc (cross multiplication).

Examples:

• 3 : 4 and 72 : 96 → Both simplify to 3 : 4 → Proportional.

• 3 : 2 and 2 : 1 → Not proportional.

Ancient Wisdom – Āryabhaṭa’s Rule of Three

Terms:

• Pramāṇa → Measure (a)

• Phala → Result (b)

• Ichchhā → Desired measure (c)

• Ichchhāphala → Required result (d)

Method:
d = (Phala × Ichchhā) ÷ Pramāṇa
→ Same as modern proportion formula.

Sharing in Ratios

To divide a quantity in a ratio m : n:

1. Add the parts (m + n).

2. Find value of one part = Total ÷ (m + n).

3. Multiply by m and n.

Examples:

• Divide 12 in 3 : 1 → 9 and 3.

• Divide 42 in 4 : 3 → 24 and 18.

• Divide 60 in 2 : 3 → 24 and 36.

Unit Conversions 

• Always use same units before comparing ratios.

Key Conversions:

• Length: 1 m = 3.281 ft

• Area: 1 m² = 10.764 ft²; 1 acre = 43,560 ft²; 1 hectare = 10,000 m² = 2.471 acres

• Volume: 1 mL = 1 cc; 1 L = 1,000 mL

Temperature:

• F = (9/5 × C) + 32

• C = 5/9 × (F − 32)

Always convert before simplifying ratios or checking proportion.