Ratio & Proportion – Advanced Study Notes
(Designed for the Sikkim PSC Quantitative Aptitude section, but framed at an MIT‑level depth)
1. Preliminaries
| Concept | Notation | Example |
|---|---|---|
| Ratio | (a:b) or (\frac{a}{b}) | 3 : 4 = (\frac{3}{4}) |
| Proportion | (a:b = c:d) | (3:4 = 6:8) |
| Unit ratio | (a:b = 1:k) | (5:10 = 1:2) |
| Unit | The second element when the first is set to 1 | In (5:10), the unit is 2 |
Remark: Ratios are ordered pairs. Swapping the elements changes the value (unless the ratio is 1:1).
2. Fundamental Properties
2.1 Equality of Ratios
Two ratios (a:b) and (c:d) are equal iff
[
a,d = b,c \tag{1}
]
Proof (cross‑multiplication):
Assume (\frac{a}{b}=\frac{c}{d}). Multiply both sides by (b,d):
[
a,d = b,c .
]
Conversely, if (a,d = b,c), divide both sides by (b,d) (assuming (b,d\neq0)):
[
\frac{a}{b} = \frac{c}{d}.
]
2.2 Transitivity
If (a:b = c:d) and (c:d = e:f), then (a:b = e:f).
Proof follows immediately from (1) by substitution.
2.3 Commensurability
If (a) and (b) are commensurable (i.e., there exists an integer (k) such that (a=kx,;b=ky) for some integers (x,y)), then the ratio is a rational number.
If they are not commensurable, the ratio is irrational.
3. Algebraic Formulation
A ratio can be represented as the ordered pair ((a,b)) with the equivalence relation
[
(a,b)\sim (c,d)\quad\Longleftrightarrow\quad a,d = b,c. \tag{2}
]
The set of all equivalence classes forms the real projective line ( \mathbb{RP}^{1}).
3.1 Proportionality as Linear Equations
The proportion (a:b = c:d) can be rewritten as
[
a,d - b,c = 0. \tag{3}
]
Thus, for a fixed (a,b), the set of all solutions ((c,d)) lies on a line in the ((c,d))-plane.
3.2 Ratios and Scalar Multiplication
For any non‑zero scalar (\lambda):
[
(a,b) = \lambda (c,d)\quad\Longleftrightarrow\quad a = \lambda c,; b = \lambda d.
]
Hence, all multiples of a ratio represent the same direction in (\mathbb{R}^2).
4. Advanced Topics
4.1 Compound Ratios
Given two successive ratios (a:b) and (b:c), their compound ratio is (a:c).
In terms of numbers:
[
\frac{a}{b}\times\frac{b}{c} = \frac{a}{c}. \tag{4}
]
Application: Scale a figure by 3:4, then by 4:5 → overall scale 3:5.
4.2 Proportion of Sums (Stewart’s Theorem for Ratios)
If (x:y = p:q) and (u:v = r:s) then
[
\frac{x+u}{y+v} = \frac{p,r + q,s}{p,s + q,r}. \tag{5}
]
Proof:
Write (\frac{x}{y}=\frac{p}{q}) and (\frac{u}{v