Ratio & Proportion – Comprehensive Study Notes
(For IIT‑JEE / Competitive Exams – Mizoram TET – Quantitative Aptitude)
Key Takeaway – Every ratio problem is essentially an algebraic equation.
Convert the word problem into a linear (or sometimes quadratic) equation, solve, then check the constraints.
1. Basic Definitions
| Symbol | Meaning |
|---|---|
| (a:b) | Ratio of (a) to (b) |
| (a:b = c:d) | Means (a,b,c,d) are in proportion |
| (\frac{a}{b}=k) | Common ratio (k) |
2. Fundamental Properties
| Property | Formula | Example |
|---|---|---|
| Cross‑multiplication | (a:b=c:d ;\Longrightarrow; ad = bc) | If (3:4 = x:12) ⇒ (3\cdot12 = 4x \Rightarrow x=9) |
| Product of ratios | (\frac{a}{b}\cdot \frac{c}{d}= \frac{ac}{bd}) | ((2:3)\cdot(5:7)=\frac{10}{21}) |
| Ratio of sums | (\frac{a+b}{c+d} = \frac{a}{c} + \frac{b}{d}) only if (a:b=c:d) (rare) | Not generally true – beware! |
| Inversion | (\frac{a}{b} = \frac{c}{d} \iff \frac{b}{a}=\frac{d}{c}) | |
| Compound ratio | If (a:b = m:n) and (b:c = p:q) then (a:c = mp:nq) |
3. Solving Ratio Problems – Step‑by‑Step
- Identify the unknowns – Usually one or two variables.
- Translate the word statement into equations
- Use cross‑multiplication for direct proportion.
- Use linear combinations for “sum” or “difference” constraints.
- Solve the system – algebraic manipulation, substitution or elimination.
- Check the bounds – Many problems restrict the value of the unknowns (e.g., integers, non‑negative, etc.).
3.1 Example 1 (Simple Proportion)
If 5 apples cost ₹30 and 8 apples cost ₹?
Let (x) be the cost of 8 apples.
[
\frac{5}{30} = \frac{8}{x} ;\Longrightarrow; 5x = 240 ;\Rightarrow; x=48
]
3.2 Example 2 (Ratio in a Sum)
The ratio of the sum of two numbers to their product is 3:4. One number is 2 more than the other. Find the numbers.
Let numbers be (x) and (x+2).
[
\frac{x+(x+2)}{x(x+2)} = \frac{3}{4}
]
[
\frac{2x+2}{x^2+2x} = \frac{3}{4}
]
Cross‑multiply:
[
4(2x+2)=3(x^2+2x)
]
[
8x+8 = 3x^2+6x
]
[
3x^2-2x-8=0
]
Solve quadratic: (x = 2) or (x = -4/3).
Take positive integer → (x=2), other number (=4).
4. Advanced Techniques
| Technique | When to Use | Key Idea |
|---|---|---|
| Substitution Method | One equation linear, the other quadratic | Solve for one variable, substitute |
| Balancing Method | Two ratios with a common term | Multiply by LCM to eliminate denominators |
| Using Bounds & Constraints | When solutions must be integers or in a range | Test integer values or use inequalities |
| Compound Ratio | Two successive proportions | Multiply numerators and denominators |
| Transforming Ratios | When ratio involves sums/differences | Express sums/differences in terms of variables |
5. Tricky Points & Common Mistakes
| Mistake | Why It Happens | Fix |
|---|---|---|
| Assuming (a+b) ratio equals (\frac{a}{b}) | Misreading “ratio of sums”. | Always set up equation: (\frac{a+b}{c+d}). |
| Ignoring integer constraints | Forgetting that “ratio” often implies whole numbers. | After solving, check if the solution satisfies all constraints. |
| Mis‑applying cross‑multiplication | Using (a:b=c:d) when the problem states “difference” or “sum”. | Translate “difference” into (a-b) or ( |
| Not simplifying fractions | Leads to large numbers that obscure the solution. | Reduce fractions early; use GCD. |
| Confusing “ratio” with “percentage” | Treating 3:4 as 75% vs 25% incorrectly. | Remember (3:4 = 3/7) of the total, not 75% of one part. |