Ratio and Proportion – Complete Guide
Exam Level: Sainik School | Content Type: Full Chapter Notes + Video Classes
1️⃣ Ratio
The ratio of two quantities a and b (same units) is:
Ratio = a / b = a : b
- a = Antecedent
- b = Consequent
Example: 5 : 9 → Antecedent = 5, Consequent = 9
Rule of Ratios
Multiplying or dividing both terms by the same non-zero number does not change the ratio.
4 : 5 = 8 : 10 = 12 : 15
4 : 6 = 2 : 3
2️⃣ Proportion
If two ratios are equal, they are in proportion:
a : b = c : d ⇒ a : b :: c : d
- a & d = Extremes
- b & c = Means
Product Rule: a × d = b × c
3️⃣ Special Proportionals
Fourth Proportional
If a : b = c : d, then d is fourth proportional.
Third Proportional
If a : b = b : c, then c is third proportional.
Mean Proportional
Mean proportional between a and b = √(a × b)
4️⃣ Compound & Multiple Ratios
Compound Ratio of (a:b), (c:d), (e:f) = ace : bdf
Duplicate Ratio = a² : b²
Sub-duplicate Ratio = √a : √b
Triplicate Ratio = a³ : b³
Sub-triplicate Ratio = ∛a : ∛b
5️⃣ Variations
Direct Proportion
x ∝ y ⇒ x = ky
Inverse Proportion
x ∝ 1/y ⇒ xy = k
6️⃣ Importance for Sainik School Exam 🎯
- Frequently asked topic in Mathematics.
- Important for Speed–Time–Distance.
- Age Problems.
- Work and Wages.
- Mixture & Alligation.
- Profit & Loss.
