Ratio and Proportion — Revision Notes

⚡ 30-Second Revision

Ratio (a:b): — Comparison of two quantities. a = antecedent, b = consequent.
Proportion (a:b :: c:d): — Equality of two ratios. ad = bc.
Compound Ratio: — (a*c):(b*d) for a:b and c:d.
Duplicate Ratio: — a²:b².
Sub-duplicate Ratio: — √a:√b.
Triplicate Ratio: — a³:b³.
Sub-triplicate Ratio: — ³√a:³√b.
Mean Proportional: — b = √ac (for a:b::b:c).
Third Proportional: — c = b²/a (for a:b::b:c).
Fourth Proportional: — d = bc/a (for a:b::c:d).
Direct Proportion: — x/y = k (constant).
Inverse Proportion: — x*y = k (constant).
Partnership Profit: — P1:P2 = (C1*T1):(C2*T2).
Mixture (Alligation): — Ratio of quantities = (C2-Cm):(Cm-C1).

2-Minute Revision

Let's quickly revise with a common problem type: Combining Ratios.

Problem: If A:B = 2:3 and B:C = 4:5, find A:B:C.

Step-by-step Approach:

1
Identify the common term: — Here, 'B' is common in both ratios.
2
Find the LCM of the common term's values: — In A:B (2:3), B is 3. In B:C (4:5), B is 4. The LCM of 3 and 4 is 12.
3
Adjust the first ratio: — To make B = 12, multiply A:B (2:3) by 4. So, A:B = (2*4):(3*4) = 8:12.
4
Adjust the second ratio: — To make B = 12, multiply B:C (4:5) by 3. So, B:C = (4*3):(5*3) = 12:15.
5
Combine the ratios: — Now that B is consistent (12 in both), we can write A:B:C = 8:12:15.

Vyyuha Insight: This method ensures the proportional relationship is maintained across all three quantities. This is a fundamental skill for solving more complex problems involving multiple variables. Always ensure the common term is unified before combining ratios.

5-Minute Revision

Let's tackle a more complex problem integrating multiple concepts:

Problem: A vessel contains 60 liters of milk and water in the ratio 3:2. If 'x' liters of the mixture are removed and 'x' liters of water are added, the ratio of milk to water becomes 1:1. Find the value of 'x'.

Step-by-step Approach:

1
Initial quantities: — Total mixture = 60 liters. Ratio Milk:Water = 3:2. Total ratio units = 3+2 = 5.

* Initial Milk = (3/5) * 60 = 36 liters. * Initial Water = (2/5) * 60 = 24 liters.

1
Removing 'x' liters of mixture: — When 'x' liters of mixture are removed, milk and water are removed in their original ratio (3:2).

* Milk removed = (3/5) * x liters. * Water removed = (2/5) * x liters.

1
Quantities after removal:

* Milk remaining = 36 - (3x/5) liters. * Water remaining = 24 - (2x/5) liters.

1
Adding 'x' liters of water: — Only water is added.

* Final Milk = 36 - (3x/5) liters. * Final Water = 24 - (2x/5) + x liters = 24 + (3x/5) liters.

1
New ratio: — The final ratio of milk to water is 1:1, meaning Final Milk = Final Water.

* 36 - (3x/5) = 24 + (3x/5)

1
Solve for 'x':

* 36 - 24 = (3x/5) + (3x/5) * 12 = (6x/5) * 6x = 12 * 5 * 6x = 60 * x = 10 liters.

Vyyuha Insight: This problem combines initial ratio distribution, removal of mixture (maintaining the ratio), addition of a single component, and then solving an equation based on the new ratio. The key is to track the quantities of milk and water separately at each stage. Such multi-step problems are common in CSAT, testing your ability to apply ratio concepts sequentially and accurately. Always ensure you're consistent with units and what's being added or removed.

Prelims Revision Notes

For Prelims, a rapid recall of Ratio and Proportion fundamentals is essential. Remember that a ratio is a comparison (a:b), while a proportion is an equality of two ratios (a:b :: c:d). The product of extremes equals the product of means (ad=bc) is your go-to formula for proportions.

Quickly identify types of ratios: compound (multiply terms), duplicate (square terms), triplicate (cube terms), and their sub-forms (square/cube roots). For mean proportional (b=√ac), third proportional (c=b²/a), and fourth proportional (d=bc/a), memorize the formulas.

Distinguish clearly between direct (x/y=k) and inverse (xy=k) proportions; this is critical for Time & Work or Speed problems. In partnership problems, profit is shared based on the product of capital and time (C*T).

For mixture and alligation, the rule of alligation (ratio of quantities = (C2-Cm):(Cm-C1)) is a major time-saver. Age problems typically involve setting up linear equations using 'x' as the common multiplier for ratios.

Always simplify ratios to their lowest terms. Practice combining ratios (e.g., A:B and B:C to A:B:C) by making the common term equal. Focus on speed and accuracy by using the 'unit method' or 'k-method' to represent unknown quantities.

Avoid common traps like incorrect unit conversions or misinterpreting 'more than'/'less than' in ratio contexts. Regular practice with PYQs will solidify these concepts and build exam-day confidence.

Mains Revision Notes

For Mains, Ratio and Proportion is less about direct problem-solving and more about conceptual application and interpretation in broader contexts. Revision should focus on understanding how proportional reasoning underpins various aspects of governance, economics, and social analysis.

Recall how ratios are used to define and evaluate economic indicators: fiscal deficit as a ratio of GDP, debt-to-GDP ratio, investment-to-GDP ratio, and current account deficit as a percentage of GDP.

Understand their significance for economic health, policy decisions, and international comparisons. In Polity, revisit proportional representation systems (e.g., Rajya Sabha elections) and how they ensure equitable representation based on population or vote share.

In Geography and Sociology, think about demographic ratios like sex ratio, dependency ratio, and population density, and their implications for social planning and resource allocation. The key is to move beyond numerical calculations and critically analyze what these ratios *mean* in a real-world context.

How do changes in these ratios reflect societal trends or policy impacts? What are the limitations of relying solely on ratios for decision-making? For instance, a high debt-to-GDP ratio might be alarming for a developed economy but acceptable for a developing one investing in infrastructure.

Practice articulating these insights in essay format, connecting quantitative data to qualitative analysis. This analytical framework, as highlighted by Vyyuha Connect, is what distinguishes a mere calculator from a future administrator capable of informed decision-making.

Vyyuha Quick Recall

Vyyuha Quick Recall: Use the 'RAPID' Mnemonic for Ratio and Proportion

R - Relationships: Always identify if it's a Direct (x/y=k) or Inverse (xy=k) relationship first. This sets the foundation. A - Alligation & Ages: Remember the Rule of Alligation for mixtures and the 'x' method for age problems.

These are high-yield areas. P - Partnerships & Proportions: For partnerships, profit is (Capital × Time). For proportions, 'Product of Extremes = Product of Means' (ad=bc) is paramount. Also recall Mean, Third, Fourth proportionals.

I - Integration & Types: Think about how ratios integrate with Percentages, Averages, Time & Work. Recall different ratio types: Compound, Duplicate, Triplicate and their sub-forms. D - Deduction & Simplification: Always simplify ratios to their lowest terms.

Deduce unknown values using the 'unit method' or 'k-method' for efficiency. Practice pattern identification to quickly solve problems.