Ratio and Proportion
Overview
More importantly, it underpins several other topics—Partnership, Mixture and Alligation, Time and Work, and even some Data Interpretation problems. Mastering this topic gives you quick marks and builds calculation speed for related areas.
At the Clerk level, questions are straightforward: expect direct ratio problems, simple proportion applications, and basic compound ratio calculations. The examiner tests whether you can manipulate ratios quickly and apply proportionality to real-world scenarios like income division, age comparisons, or quantity distribution. Speed and accuracy matter more than complexity here.
Key Concepts
- Ratio is a comparison of two quantities by division. If A : B = 3 : 4, it means for every 3 units of A, there are 4 units of B. Ratios have no units—they are pure numbers.
- Proportion states that two ratios are equal. If A : B = C : D, then A, B, C, D are in proportion. This is written as A : B :: C : D.
- The "k-method": When A : B = 3 : 4, assume A = 3k and B = 4k, where k is a common multiplier. This converts ratios into actual values for calculation.
- Compound Ratio: The compound ratio of (a : b) and (c : d) is (a × c) : (b × d). Used when two ratios combine multiplicatively.
- Duplicate and Triplicate Ratios: Duplicate ratio of a : b is a² : b². Triplicate ratio is a³ : b³. Sub-duplicate is √a : √b.
- Componendo and Dividendo: If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d). This shortcut appears occasionally in Clerk-level problems.
- Direct Proportion: When two quantities increase or decrease together at the same rate (e.g., more items, more cost).
- Inverse Proportion: When one quantity increases while the other decreases proportionally (e.g., more workers, less time).
Formulas / Key Facts
| Formula/Rule | Context |
|---|---|
| If A : B = a : b, then A = ak, B = bk | Converting ratio to values |
| Mean Proportional of a and b = √(a × b) | Middle term when a : x = x : b |
| Third Proportional to a, b = b²/a | When a : b = b : x |
| Fourth Proportional to a, b, c = (b × c)/a | When a : b = c : x |
| Product of Extremes = Product of Means | a × d = b × c in a : b :: c : d |
| Compound Ratio of (a : b) and (c : d) = ac : bd | Multiplying ratios |
| If A : B = 2 : 3 and B : C = 4 : 5, make B common first | Combining ratios |
| Division in ratio a : b : total T gives a/(a+b) × T | Dividing quantities |
Worked Examples
Example 1: Basic Ratio Division
Problem: Rs. 630 is divided between A and B in the ratio 4 : 5. Find A's share.
Solution:
- Total parts = 4 + 5 = 9
- A's share = (4/9) × 630 = 280
- Answer: Rs. 280
Example 2: Combining Two Ratios
Problem: If A : B = 2 : 3 and B : C = 5 : 7, find A : B : C.
Solution:
- B appears in both ratios. Make B's value equal.
- In first ratio, B = 3. In second ratio, B = 5.
- LCM of 3 and 5 = 15
- Multiply first ratio by 5: A : B = 10 : 15
- Multiply second ratio by 3: B : C = 15 : 21
- A : B : C = 10 : 15 : 21
Example 3: Finding Original Quantities
Problem: The ratio of two numbers is 3 : 5. If 9 is added to each number, the ratio becomes 3 : 4. Find the numbers.
Solution:
- Let numbers be 3k and 5k
- After adding 9: (3k + 9)/(5k + 9) = 3/4
- Cross multiply: 4(3k + 9) = 3(5k + 9)
- 12k + 36 = 15k + 27
- 3k = 9, so k = 3
- Numbers are 3 × 3 = 9 and 5 × 3 = 15
- Answer: 9 and 15
Example 4: Compound Ratio
Problem: Find the compound ratio of 2 : 3, 5 : 7, and 4 : 9.
Solution:
- Compound ratio = (2 × 5 × 4) : (3 × 7 × 9)
- = 40 : 189
- Answer: 40 : 189
Common Mistakes
- Adding instead of using the k-method → When ratio is 3 : 4, students sometimes assume actual values are 3 and 4. Always use 3k and 4k, then find k from given conditions.
- Forgetting to make common terms equal when combining ratios → For A : B = 2 : 3 and B : C = 4 : 5, directly writing A : B : C = 2 : 3 : 5 is wrong. Make B equal first using LCM.
- Confusing ratio with actual difference → If A : B = 5 : 3, the difference is 2 parts, not 2 units. Calculate actual difference as 2k.
- Misapplying proportion direction → In inverse proportion problems (like time-work), students sometimes multiply instead of dividing. Ask yourself: "If one increases, does the other increase or decrease?"
- Ignoring simplification → Always express final ratios in lowest terms. 12 : 18 should be written as 2 : 3.
Quick Reference
- Ratio a : b means values are ak and bk — find k from additional information.
- To combine A : B and B : C, equalize B using LCM, then merge.
- Fourth proportional to a, b, c = bc/a — quick formula for proportion problems.
- Compound ratio = multiply all antecedents : multiply all consequents.
- "Increased by ratio" problems: new value = original × (new ratio term/old ratio term).
- Check: In proportion a : b :: c : d, always verify a × d = b × c.