Ratio Proportion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ratio Proportion — Quick Facts for SSC CGL

Ratio: A ratio is a comparison of two quantities of the same kind, expressed in the same units. It is written as a:b, where a and b are non-zero numbers. a:b = a/b.

Key Properties:

• a:b = ka:kb for any non-zero k
• The ratio should be simplified to its lowest terms
• If a:b = c:d, then a/b = c/d → ad = bc

Types of Ratios:

• Compound Ratio: (a:b) × (c:d) = (ac : bd)
• Duplicate Ratio: (a:b) = (a² : b²)
• Sub-duplicate Ratio: (a:b) = (√a : √b)
• Reciprocal Ratio: (a:b) = (b : a) for a and b both non-zero

Division in a Given Ratio: If a quantity Q is divided in the ratio a:b, then:

• Part 1 = (a/(a+b)) × Q
• Part 2 = (b/(a+b)) × Q

For division in ratio a:b:c:

• Part 1 = (a/(a+b+c)) × Q
• Part 2 = (b/(a+b+c)) × Q
• Part 3 = (c/(a+b+c)) × Q

Proportion: Four quantities a, b, c, d are in proportion if a:b = c:d. Written as a:b :: c:d. Here, a and d are called extremes; b and c are called means. If a:b = c:d → ad = bc

Continued Proportion: a, b, c are in continued proportion if a:b = b:c → b² = ac (b is the mean proportional between a and c; c is the third proportional to a and b)

⚡ Exam tip: If A:B = 3:5 and B:C = 4:7, then A:B:C = ? Multiply the ratios: A:B = 3:5 and B:C = 4:7. Make B equal: LCM of 5 and 4 = 20. So A:B = 12:20 and B:C = 20:35 → A:B:C = 12:20:35.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Ratio Proportion — SSC CGL Study Guide

Direct and Indirect Proportion:

Type	Relation	Example
Direct Proportion	When one increases, other also increases (x ∝ y)	More articles → more cost (at same price)
Indirect/Inverse Proportion	When one increases, other decreases (x ∝ 1/y)	More workers → less time to complete job

Rule of Three:

Direct: If a quantity changes in the same ratio as another: A is to B as C is to D → A/B = C/D → D = BC/A

Inverse: If the product of two quantities is constant: A is to B as C is to D (inverse) → A × B = C × D

Chain Rule: Used for problems involving multiple proportional relationships: Example: If 8 machines produce 200 units in 5 days, how many units will 12 machines produce in 10 days? Solution: More machines → more output (direct) More days → more output (direct) 12/8 × 10/5 × 200 = 600 units

Partnership: When two or more persons invest together:

• When investments are for the same time: Profit/Loss is divided in ratio of investments
• When investments are for different time periods: (Investment × Time) is the determining factor

Example: A invests ₹30,000 for 8 months, B invests ₹40,000 for 6 months. If profit is ₹14,700, how is it divided?

• A’s contribution = 30000 × 8 = 2,40,000
• B’s contribution = 40000 × 6 = 2,40,000
• Ratio = 2,40,000 : 2,40,000 = 1 : 1
• A gets = 14700 × 1/2 = ₹7,350
• B gets = ₹7,350

Mixture and Alligation:

Alligation Rule: When two ingredients of different prices are mixed:

• Mean price = (Quantity of cheaper × Price of cheaper + Quantity of expensive × Price of expensive) / Total quantity

Alligation Method: If the mean price lies between the prices of two ingredients:

Cheaper (₹p₁)	Mean Price (₹m)	Costlier (₹p₂)
p₂ − m		m − p₁

Ratio of quantities = (p₂ − m) : (m − p₁)

Example: In what ratio must wheat at ₹20/kg be mixed with wheat at ₹30/kg to get a mixture costing ₹24/kg?

• p₁ = 20, p₂ = 30, m = 24
• p₂ − m = 30 − 24 = 6
• m − p₁ = 24 − 20 = 4
• Ratio = 6 : 4 = 3 : 2

⚡ Common mistake: In mixture problems, students sometimes use simple average instead of alligation. Remember: alligation gives weighted average — the actual mean price must lie between the two ingredient prices.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Ratio Proportion — Comprehensive Notes

Advanced Ratio Problems:

Example 1 (Income and Expenditure): A person’s income and expenditure are in the ratio 5:4. His income increases by 20% and expenditure by 30%. By what percentage does his savings increase?

• Original: Income = 5x, Expenditure = 4x, Savings = x
• New Income = 5x × 1.2 = 6x
• New Expenditure = 4x × 1.3 = 5.2x
• New Savings = 6x − 5.2x = 0.8x
• Increase in savings = (0.8x − x)/x × 100 = −20%
• Savings DECREASED by 20% (not increased!)

Example 2 (Working Together): A can complete a work in 20 days, B can complete it in 30 days. How many days will they take working together?

• A’s 1 day work = 1/20
• B’s 1 day work = 1/30
• Combined 1 day work = 1/20 + 1/30 = (3+2)/60 = 5/60 = 1/12
• Days to complete = 12 days

Example 3 (Pipes and Cisterns — analogous to work): A pipe fills a tank in 20 minutes, another empties it in 30 minutes. How long to fill the tank?

• Fill rate = 1/20 per minute
• Empty rate = 1/30 per minute
• Net fill rate = 1/20 − 1/30 = 1/60
• Time to fill = 60 minutes

Example 4 (Changing Ratios): A sum of ₹2,000 is divided among A, B, C in the ratio 2:3:5. If A’s share becomes ₹800 after receiving some amount from C, find how much C gave to A.

• Original: A = 2x, B = 3x, C = 5x; 2x + 3x + 5x = 2000 → 10x = 2000 → x = 200
• A’s original share = ₹400; B’s = ₹600; C’s = ₹1000
• A’s new share = ₹800
• A received ₹400 from C
• C gave ₹400 → C’s new share = ₹600
• New ratio = 800:600:600 = 4:3:3

Example 5 (Age Problems with Ratios): Five years ago, the ratio of ages of A and B was 3:4. Five years hence, it will be 5:6. Find their present ages.

• 5 years ago: A = 3k, B = 4k
• Present: A = 3k+5, B = 4k+5
• 5 years hence: A = 3k+10, B = 4k+10
• (3k+10)/(4k+10) = 5/6
• 6(3k+10) = 5(4k+10)
• 18k + 60 = 20k + 50
• 10 = 2k → k = 5
• Present ages: A = 20 years, B = 25 years

Continued Proportion — Third Proportional: If a:b = b:c, then c is the third proportional to a and b. Example: Find the third proportional to 4 and 6.

• b² = ac → 6² = 4 × c → 36 = 4c → c = 9

Mean Proportional: Mean proportional between a and c = √(ac) Example: Mean proportional between 3 and 12 = √(3×12) = √36 = 6

NEET/SSC Pattern Analysis: Ratio and Proportion is a frequently tested topic in SSC CGL Tier-II (Quantitative Aptitude). Key areas: dividing amounts in given ratios, direct and inverse proportion word problems, partnership profit-sharing, alligation (mixtures), and age problems. Pipe and cistern problems are also essentially ratio-based.

⚡ SSC CGL 2022 Qn: If A:B = 2:3 and B:C = 4:5, then A:C is equal to: Answer: A:C = 8:15 (LCM of 3 and 4 is 12; A:B = 8:12, B:C = 12:15 → A:C = 8:15).