Ratio & Proportion – Study Notes
(For Puducherry Forest Guard – Quantitative Aptitude)
1. Simple Explanation
| Term | Meaning | Symbol |
|---|---|---|
| Ratio | A comparison of two numbers showing how many times one value contains the other | (a:b) or (\frac{a}{b}) |
| Proportion | An equation stating that two ratios are equal | (\frac{a}{b}=\frac{c}{d}) |
Key Idea
If the ratio of (A) to (B) is the same as the ratio of (C) to (D), we can set up a proportion and solve for any unknown using cross‑multiplication.
2. Key Points & Formulae
| Concept | Formula | Notes |
|---|---|---|
| Direct Variation | (y = kx) | Constant (k) = (\frac{y}{x}). Increase in (x) → proportional increase in (y). |
| Inverse Variation | (y = \frac{k}{x}) | Constant (k) = (xy). Increase in (x) → proportional decrease in (y). |
| Cross Multiplication | If (\frac{a}{b}=\frac{c}{d}) then (ad = bc) | Used to find any missing term. |
| Compound Ratio | If (A:B = m:n) and (B:C = p:q) then (A:C = m\cdot p : n\cdot q) | Useful for multi‑step ratio problems. |
| Adding/Subtracting Ratios | (A:B = m:n) and (C:D = p:q) → sum = ((m+p):(n+q)) | Only when the same unit is being compared. |
| Percentage from Ratio | (\frac{part}{whole} = \text{ratio}) → (\text{percentage} = \frac{\text{ratio}}{whole}\times100%) | Convert (a:b) to percentage of (b). |
| Scale Factor | If two similar figures are scaled by (k), all linear dimensions multiply by (k). | Used in area & volume scaling. |
3. Types of Ratio & Proportion Problems
| Type | Typical Question | Approach |
|---|---|---|
| Direct Proportion | “If 2 workers finish a job in 5 days, how many days for 5 workers?” | Set up (\frac{\text{time}}{\text{workers}}) = constant. |
| Inverse Proportion | “A machine takes 10 hrs to produce 200 units. How long for 140 units?” | Use ( \text{time}\times\text{units} = k). |
| Compound Ratio | “The ratio of A:B is 3:4 and B:C is 2:5. Find A:C.” | Multiply numerators & denominators. |
| Mixture Problems | “Blend 3 L of 20% salt and 7 L of 30% salt. What is final concentration?” | Use weighted averages. |
| Scaling Problems | “Scale a map 1 : 1000 to 1 : 500.” | Double lengths, quadruple area. |
| Ratio with Parts | “A:b = 4:5 and a+b = 45. Find a and b.” | Set (a = 4k, b = 5k; a+b=9k=45). |
| Percentage to Ratio | “15% of a number is 18. Find the number.” | (0.15x = 18 \Rightarrow x=120). |
4. Worked Examples
Example 1 – Direct Proportion
Problem
A team of 6 workers can complete a forest patrol in 12 days. How many days will it take 9 workers to complete the same patrol, keeping all other conditions same?
Solution
Let (T) be time.
Direct proportion: (\frac{T}{\text{workers}} = \text{constant}).
[ \frac{12}{6} = \frac{T}{9} ]
Cross‑multiply:
(12 \times 9 = 6T \Rightarrow 108 = 6T \Rightarrow T = 18) days.
Answer: 18 days.
Example 2 – Inverse Proportion
Problem
A forest guard vehicle can travel 240 km in 6 hrs. How many hours