Ratio & Proportion – Study Notes
Delhi Patwari / Lekhpal – Quantitative Aptitude
1. Simple Explanation
| Term | Meaning |
|---|---|
| Ratio | A comparison of two or more numbers expressed as a fraction, e.g. (a:b = \frac{a}{b}). |
| Proportion | An equation that states two ratios are equal, e.g. (\frac{a}{b} = \frac{c}{d}) or (a:b = c:d). |
Key Idea – If two ratios are equal, the cross‑products are equal:
[
a:b = c:d ;;\Longrightarrow;; a \times d = b \times c
]
2. Key Points
| Concept | How to Use | Common Pitfall |
|---|---|---|
| Direct & Inverse Proportions | - Direct: (x \propto y) → (x = k \cdot y) - Inverse: (x \propto \frac{1}{y}) → (x = \frac{k}{y}) |
Mixing up direct vs inverse can lead to wrong proportionality constant. |
| Finding a Missing Term | Use cross‑multiplication. | Forgetting to set up the equation correctly. |
| Compound Ratio | Multiply successive ratios. | Forgetting to reduce to simplest form. |
| Percentage from Ratio | If (a:b = p:q) and (a) is (p%) of total, total = (a \times \frac{100}{p}). | Misinterpreting “percentage” as a fraction of a whole. |
| Unit Price / Cost | Unit Price = Cost ÷ Quantity. | Using money symbols interchangeably with numbers. |
| Ratios in Word Problems | Translate “for every 3 apples, there are 5 oranges” → (3:5). | Ignoring the “for every” part; converting to fractions incorrectly. |
3. Worked Example
Problem
A land is divided into three parts in the ratio 4 : 5 : 6.
If the total area is 360 ha, find the area of each part.
Solution
-
Sum of ratio parts
[ 4 + 5 + 6 = 15 ] -
Value of one part (unit area)
[ \text{One part} = \frac{\text{Total area}}{\text{Sum of parts}} = \frac{360}{15} = 24 \text{ ha} ] -
Individual areas
- First part: (4 \times 24 = 96) ha
- Second part: (5 \times 24 = 120) ha
- Third part: (6 \times 24 = 144) ha
Check – (96 + 120 + 144 = 360) ha ✔️
4. Quick Revision Summary
- Ratio: (a:b = \frac{a}{b})
- Proportion: (\frac{a}{b} = \frac{c}{d}) → (a \times d = b \times c)
- Direct Proportion: (x = k \cdot y)
- Inverse Proportion: (x = \frac{k}{y})
- Compound Ratio: Multiply ratios step‑by‑step.
- One‑part value: (\frac{\text{Total}}{\text{Sum of parts}})
- Cross‑multiplication: Solve for missing term.
- Word‑to‑ratio: Translate “for every X, there are Y” → (X:Y).
- Check: Sum of parts × one‑part value = total.
Use this checklist before tackling any ratio & proportion problem to avoid common mistakes. Happy practicing!