Ratio and Proportion
Overview
Ratio and Proportion is a foundational arithmetic topic that appears consistently in SBI Clerk Prelims, either as direct questions or embedded within Data Interpretation, Partnership, and Mixture problems.
The topic tests your ability to compare quantities, scale values proportionally, and solve multi-step problems involving compound ratios. Mastering ratio and proportion gives you speed advantages across the Numerical Ability section since these concepts underpin profit-sharing, mixture calculations, and even time-work problems.
For SBI Clerk Prelims, focus on quick mental calculations with simple ratios, compound ratio formation, and proportion-based equation solving. Problems are calculation-intensive but conceptually straightforward—accuracy and speed matter more than complex reasoning.
Key Concepts
- Ratio expresses the relative size of two quantities. If A:B = 3:4, it means for every 3 units of A, there are 4 units of B. Ratios have no units and can be scaled by any common multiplier.
- Proportion states that two ratios are equal. If A:B = C:D, then A×D = B×C (cross-multiplication rule). This is your primary solving tool.
- Compound Ratio is found by multiplying corresponding terms. If ratios are a:b and c:d, their compound ratio is ac:bd. Used when quantities depend on multiple factors.
- Duplicate and Triplicate Ratios: Duplicate of a:b is a²:b². Triplicate is a³:b³. Sub-duplicate is √a:√b.
- Componendo-Dividendo: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Useful for quickly solving certain proportion equations.
- Distribution in Ratio: To divide quantity Q in ratio a:b:c, the parts are Qa/(a+b+c), Qb/(a+b+c), Qc/(a+b+c).
- Combining Ratios: When A:B = 2:3 and B:C = 4:5, make B common (LCM of 3 and 4 = 12), so A:B:C = 8:12:15.
- Inverse Ratio: If A:B = 3:4, then inverse ratio is 4:3. Used when quantities are inversely related (like speed and time for same distance).
Formulas / Key Facts
Basic Ratio: If A:B = a:b, then A = ak and B = bk for some constant k.
Proportion Rule: If a:b :: c:d, then ad = bc (product of extremes = product of means).
Compound Ratio: (a:b) compounded with (c:d) = ac:bd.
Duplicate Ratio of a:b = a²:b².
Sub-duplicate Ratio of a:b = √a:√b.
Triplicate Ratio of a:b = a³:b³.
Division Formula: Share of A when amount M is divided in ratio a:b:c is M × a/(a+b+c).
Combining Two Ratios: A:B = p:q and B:C = r:s → A:B:C = pr : qr : qs (after making B equal).
Mean Proportional of a and b = √(ab). If a:x :: x:b, then x = √(ab).
Third Proportional to a and b: If a:b :: b:x, then x = b²/a.
Worked Examples
Example 1: Basic Distribution
₹2400 is divided among A, B, and C in the ratio 3:5:4. Find each person's share.
Solution:
- Sum of ratio parts = 3 + 5 + 4 = 12
- A's share = 2400 × 3/12 = 2400 × 1/4 = ₹600
- B's share = 2400 × 5/12 = ₹1000
- C's share = 2400 × 4/12 = ₹800
Example 2: Combining Ratios
If A:B = 2:3 and B:C = 5:7, find A:B:C.
Solution:
- B appears as 3 in first ratio and 5 in second
- LCM of 3 and 5 = 15
- Multiply first ratio by 5: A:B = 10:15
- Multiply second ratio by 3: B:C = 15:21
- Combined: A:B:C = 10:15:21
Example 3: Compound Ratio Application
The incomes of P and Q are in ratio 4:3. Their expenditures are in ratio 3:2. If each saves ₹1000, find their incomes.
Solution:
- Let incomes be 4x and 3x
- Let expenditures be 3y and 2y
- Savings: Income − Expenditure = 1000
- For P: 4x − 3y = 1000
- For Q: 3x − 2y = 1000
- From both equations: 4x − 3y = 3x − 2y
- Solving: x = y
- Substituting in first equation: 4x − 3x = 1000, so x = 1000
- P's income = 4 × 1000 = ₹4000
- Q's income = 3 × 1000 = ₹3000
Example 4: Finding Original Ratio
Two numbers are in ratio 3:4. If 5 is added to each, the ratio becomes 4:5. Find the numbers.
Solution:
- Let numbers be 3k and 4k
- After adding 5: (3k + 5)/(4k + 5) = 4/5
- Cross-multiply: 5(3k + 5) = 4(4k + 5)
- 15k + 25 = 16k + 20
- k = 5
- Numbers are 15 and 20
Common Mistakes
Adding/subtracting ratios directly → Wrong. 2:3 and 4:5 don't add to 6:8. Ratios must be combined using LCM method or compounding rules.
Forgetting the constant multiplier → When ratio is 3:4, actual values are 3k and 4k, not 3 and 4. Always introduce 'k' for calculations.
Confusing compound ratio with addition → Compound of 2:3 and 4:5 is 8:15, not 6:8. Multiply corresponding terms.
Misplacing terms in proportion → In a:b :: c:d, ad = bc. Students often multiply wrong pairs. Remember: extremes (first and last) multiply together.
Not simplifying final ratios → Answer 12:18 should be written as 2:3. Always reduce to lowest terms unless question specifies otherwise.
Quick Reference
- Ratio a:b means actual values are ak and bk—always use the multiplier k.
- Proportion: If a:b = c:d, then ad = bc (cross-multiply).
- Compound ratio: Multiply term-by-term (ac:bd).
- To combine A:B and B:C, make B common using LCM, then merge.
- Distribution: Part = Total × (own ratio term / sum of all terms).
- Mean proportional of a and b = √(ab).