Ratio & Proportion – Advanced Quantitative Aptitude Notes
(MIT‑style, first‑principles approach with rigorous reasoning, edge‑cases & a challenging problem)
1. What Is a Ratio?
| Symbol | Definition |
|---|---|
| (a:b) | The ratio of (a) to (b) is the ordered pair ((a,b)) or the real number (\displaystyle \frac{a}{b}) when (b\neq 0). |
| (\frac{a}{b}) | The quotient (a) divided by (b). |
Key Points
- The ratio is ordered; (3:4\neq 4:3).
- When (b=0) the ratio is undefined (edge‑case).
- Ratios can be positive, negative, or zero (if (a=0)).
2. Proportion
A proportion is an equality between two ratios:
[ \frac{a}{b} = \frac{c}{d}\quad \Leftrightarrow\quad a:b = c:d ]
with (b,d \neq 0).
Cross‑multiplication follows immediately:
[ ad = bc \quad \text{(provided } b,d\neq 0\text{)}. ]
3. From First Principles
3.1 Ratio as an Element of the Projective Line
In projective geometry, the ratio (a:b) is an equivalence class under scalar multiplication:
[ (a,b) \sim (\lambda a,\lambda b), \qquad \lambda \in \mathbb{R}\setminus{0}. ]
Two ordered pairs represent the same ratio iff they are proportional.
This viewpoint gives a clean algebraic structure:
- Set: (\mathbb{P}^1(\mathbb{R}) = {[a:b]\mid a,b\in\mathbb{R},(a,b)\neq(0,0)}).
- Addition: Not defined (since ratios are projective points).
- Multiplication: ([a:b]\cdot[c:d] = [ac:bd]).
Proof: ( (a,b)) and ( (c,d)) represent ratios; their product is given by component‑wise multiplication, which respects the equivalence relation.
3.2 Ratio Spaces in Linear Algebra
Consider a vector space (V) over (\mathbb{R}).
A ratio of two vectors (u,v\in V) is defined by
[ [u:v] = { \lambda u : \lambda v \mid \lambda\in\mathbb{R}\setminus{0}}. ]
If (u,v) are non‑zero, this is a one‑dimensional subspace of (V) (a line through the origin).
This generalises the scalar ratio to directional ratios and is fundamental in analytic geometry.
4. Algebraic Treatment – Solving Proportional Equations
Given (a:b = c:d), we often need to solve for an unknown.
General strategy:
- Cross‑multiply: (ad = bc).
- Isolate the unknown on one side.
- Simplify using algebraic identities.
4.1 Worked Example
Find (x) if
[ \frac{2x-3}{5} = \frac{9}{4x+1}. ]
Solution
[ (2x-3)(4x+1) = 5\cdot 9 = 45. ]
Expand:
[ 8x^2 + 2x - 12x - 3 = 45 ;\Rightarrow; 8x^2 - 10x - 48 = 0. ]
Divide by 2:
[ 4x^2 - 5x - 24 = 0. ]
Solve via quadratic formula:
[ x = \frac{5 \pm \sqrt{25 + 384}}{8} = \frac{5 \pm \sqrt{409}}{8}. ]
Both roots are admissible because they do not make any denominator zero.
5. Advanced Topics
5.1 Ratios in Sequences & Series
- Geometric Sequence: (a_n = ar^{n-1}).
Ratio of successive terms: (\frac{a_{n+1}}{a_n} = r). - Arithmetic Sequence: (a_n = a_1 + (n-1)d).
Difference of successive terms: (d).
Proportionality of sums: For a GP, the ratio of sums (S_m:S_n) satisfies
[ \frac{S_m}{S_n} = \frac{1-r^m}{1-r^n}, \quad r\neq 1. ]
5.2 Ratios of Functions
For functions (f,g) with (g(x)\neq 0):
[ \frac{f(x)}{g(x)} = \text{constant} ;\Rightarrow; f(x) = c,g(x). ]
If (f,g) are polynomials, this yields a proportionality condition that can be checked by comparing coefficients.
5.3 Limits & Ratios
[ \lim_{x\to a} \frac{f(x)}{g(x)} = L ]
is a ratio of limits.
If (L) is finite and non‑zero, (