Introduction
Ratio and Proportion is a direct extension of fractions — it underlies Partnership, Time-Speed-Distance
(ratio of speeds), and Mixture problems. A firm grip on the basic rules here pays off across several
later chapters.
Basic Concepts
| Term | Meaning |
|---|
| Ratio (a:b) | Comparison of two quantities of the same kind |
| Proportion (a:b :: c:d) | Equality of two ratios — a/b = c/d |
| Fourth Proportional | d, when a:b :: c:d, i.e., d = (b×c)/a |
| Third Proportional | c, when a:b :: b:c, i.e., c = b²/a |
| Mean Proportional | b, when a:b :: b:c, i.e., b = √(ac) |
Compounded Ratio
If a:b and c:d are two ratios, the compounded ratio = ac : bd (multiply corresponding terms).
Q. Find the compounded ratio of 2:3 and 4:5.
Compounded ratio = (2×4) : (3×5) = 8:15
Direct vs Inverse Proportion
Flowchart — Two Types of Proportion
Direct Proportion — both quantities increase/decrease together (e.g., cost ∝ quantity)
Inverse Proportion — one increases as the other decreases (e.g., speed ∝ 1/time for fixed distance)
Dividing a Quantity in a Given Ratio
Q. Divide ₹4,500 among A, B, and C in the ratio 2:3:4.
Total parts = 2+3+4 = 9
A = (2/9)×4500 = ₹1000, B = (3/9)×4500 = ₹1500, C = (4/9)×4500 = ₹2000
Changing Ratios — "Add/Subtract to Equalise" Pattern
Q. The ratio of two numbers is 3:5. If 10 is added to each, the ratio becomes 5:7. Find the numbers.
Let numbers = 3x, 5x
(3x+10)/(5x+10) = 5/7 → 7(3x+10) = 5(5x+10) → 21x+70 = 25x+50 → 4x=20 → x=5
Numbers = 15 and 25
Mixture / Alligation Ratio Pattern
Flowchart — Alligation Rule
Ratio of quantities of cheaper : dearer = (Dearer price − Mean price) : (Mean price − Cheaper price)
Q. In what ratio should tea worth ₹60/kg be mixed with tea worth ₹90/kg to get a mixture worth ₹75/kg?
Ratio = (90−75) : (75−60) = 15 : 15 = 1:1
💡 Exam Tip: Alligation is really just a shortcut form of a weighted-average equation — recognising "mixture worth ₹X per kg" style questions instantly as alligation problems saves the need to set up algebraic equations.
✅ Practice Focus: Fourth/third/mean proportional formulas · Compounded ratio calculation · Direct vs inverse proportion identification · Alligation formula for mixture problems.