06_Ratio__Proportion___Partnership

Ratio, Proportion & Partnership - Aptitude Mastery Guide

Category: Quantitative Aptitude
Generated on: 2025-07-15 09:16:41
Source: Aptitude Mastery Guide Generator

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Ratio, Proportion, & Partnership: A Comprehensive Guide

This guide is designed to be your ultimate resource for mastering ratio, proportion, and partnership concepts. It covers the fundamentals, shortcuts, essential formulas, solved examples, and practice problems to help you excel in competitive exams and placement tests.

1. Foundational Concepts

Ratio: A ratio is a comparison of two or more quantities of the same kind. It expresses how many times one quantity contains another. We can represent the ratio of ‘a’ to ‘b’ as a:b or a/b. ‘a’ is called the antecedent, and ‘b’ is called the consequent.

• Why this matters: Ratios help us understand the relative sizes of things. Think about mixing paint – a ratio tells you how much of each color to use.

Proportion: A proportion states that two ratios are equal. If a:b = c:d, then a, b, c, and d are said to be in proportion. This can also be written as a/b = c/d. The terms ‘a’ and ‘d’ are called extremes, and ‘b’ and ‘c’ are called means.

• Why this matters: Proportions let us scale things up or down while maintaining the same relationship. Think about map reading – a scale of 1:100000 means 1 cm on the map represents 100000 cm in reality.

Partnership: A partnership is an agreement between two or more individuals to share in the profits or losses of a business. The profit or loss is usually distributed according to the ratio of their investments and the duration for which they invested.

• Why this matters: Understanding partnership is crucial for business contexts. It helps in fairly distributing profits and losses among partners based on their contributions.

Key Terms in Proportion:

• Fourth Proportional: If a:b = c:d, then ‘d’ is the fourth proportional to a, b, and c.
• Third Proportional: If a:b = b:c, then ‘c’ is the third proportional to a and b.
• Mean Proportional: The mean proportional between a and b is √(ab). This means a:x = x:b, where x is the mean proportional.

2. Key Tricks & Shortcuts

This section is the heart of this guide. It provides practical shortcuts and tricks to solve ratio, proportion, and partnership problems quickly and efficiently.

• Trick 1: Componendo and Dividendo

  • Rule: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d).
  • When to use: This is extremely useful for simplifying equations where you have sums and differences in ratios.

• Trick 2: Direct Proportion - Cross Multiplication

  • Rule: If a quantity ‘A’ is directly proportional to another quantity ‘B’, then A/B = constant. So, A₁/B₁ = A₂/B₂.
  • When to use: This is ideal for problems where one quantity increases proportionally with another (e.g., cost and number of items). Solve by cross-multiplying.

• Trick 3: Inverse Proportion – Product is Constant

  • Rule: If a quantity ‘A’ is inversely proportional to another quantity ‘B’, then A * B = constant. So, A₁ * B₁ = A₂ * B₂.
  • When to use: This is perfect for scenarios where one quantity increases as the other decreases (e.g., speed and time to cover a fixed distance).

• Trick 4: Converting Ratios to Proportions (and vice versa)

  • Rule: If A:B = x:y and B:C = p:q, then A:B:C = xp:yp:yq. Adjust the ratios to make the ‘B’ term the same.
  • When to use: When you need to combine multiple ratios to find a combined ratio.

• Trick 5: Partnership – Investment and Time Period

  • Rule: If A, B, and C invest amounts X, Y, and Z respectively for time periods P, Q, and R respectively, then the ratio of their profits is XP : YQ : Z*R.
  • When to use: This is the fundamental rule for partnership problems.

• Trick 6: Vedic Math - Base Method for Ratios (Close to a Base)

  • Concept: This adapts the Vedic Math base method for multiplication. If the ratio terms are close to a common base (like 100), find the deviations and apply a similar logic.
  • Example: Let’s say you need to find the combined ratio of A:B = 98:102 and B:C = 101:99. The base is 100.
    • A:B = -2 : +2 (deviations from 100)
    • B:C = +1 : -1 (deviations from 100)
    • Adjust B: Multiply the first ratio by 101 and the second by 102 to make the B values comparable. This becomes complex quickly, so this is only useful if the deviations are very small and the base is simple (like 10 or 100). In most cases, Trick 4 is more efficient.

• Trick 7: Percentage-to-Fraction Conversion for Ratio Simplification

  • Rule: Quickly convert percentages to fractions to simplify ratio calculations.
  • Examples:
    • 25% = 1/4
    • 33.33% (or 33 1/3%) = 1/3
    • 16.66% (or 16 2/3%) = 1/6
    • 12.5% = 1/8
  • When to use: When ratio problems involve percentages, converting them to fractions often makes calculations easier.

• Trick 8: Assumption Method (for Proportions)

  • Rule: If a proportion is given (e.g., a/b = c/d), assume a value for one variable and calculate the others based on the proportion. This simplifies calculations, especially when dealing with complex fractions.
  • When to use: When the question asks for the ratio of two quantities, rather than specific values.

3. Essential Formulas & Rules

Formula/Rule	Description
a:b = c:d => ad = bc	Product of extremes = Product of means
a:b = c:d => a/b = c/d	Alternative representation of proportion
Mean Proportional between a and b	√(ab)
Third Proportional to a and b	b²/a
Fourth Proportional to a, b, and c	(b*c)/a
Componendo & Dividendo: If a/b = c/d	(a+b)/(a-b) = (c+d)/(c-d)
Direct Proportion: A ∝ B	A/B = k (constant), or A₁/B₁ = A₂/B₂
Inverse Proportion: A ∝ 1/B	AB = k (constant), or A₁B₁ = A₂*B₂
Partnership Profit Sharing Ratio (investments only)	If A invests X and B invests Y, Profit Ratio = X:Y
Partnership Profit Sharing Ratio (investment & time)	If A invests X for P months and B invests Y for Q months, Profit Ratio = XP : YQ
Duplicate Ratio of a:b	a²:b²
Sub-duplicate Ratio of a:b	√a:√b
Triplicate Ratio of a:b	a³:b³
Sub-triplicate Ratio of a:b	³√a:³√b

4. Detailed Solved Examples

Example 1: Basic Proportion & Fourth Proportional (Using Formula)

Find the fourth proportional to 4, 9, and 12.

• Solution:
  • Let the fourth proportional be ‘x’.
  • Then, 4:9 = 12:x
  • Using the formula: x = (9 * 12) / 4 = 27
  • Therefore, the fourth proportional is 27.

Example 2: Componendo and Dividendo (Trick 1)

If (7x + 11y) : (7x - 11y) = 3:2, find the value of x:y.

• Solution:
  • Applying Componendo and Dividendo:
    • [(7x + 11y) + (7x - 11y)] / [(7x + 11y) - (7x - 11y)] = (3+2)/(3-2)
    • (14x) / (22y) = 5/1
    • (7x) / (11y) = 5/1
    • x/y = (5 * 11) / 7 = 55/7
  • Therefore, x:y = 55:7

Example 3: Partnership with Variable Investments (Trick 5)

A, B, and C start a business. A invests Rs. 12000 for 6 months, B invests Rs. 15000 for 4 months, and C invests Rs. 10000 for 9 months. At the end of the year, the total profit is Rs. 46000. Find the share of each in the profit.

• Solution:
  • Ratio of their investments multiplied by their time periods:
    • A:B:C = (12000 * 6) : (15000 * 4) : (10000 * 9)
    • A:B:C = 72000 : 60000 : 90000
    • Simplifying the ratio: A:B:C = 12:10:15
  • Total parts = 12 + 10 + 15 = 37
  • A’s share = (12/37) * 46000 = Rs. 14864.86 (approx.)
  • B’s share = (10/37) * 46000 = Rs. 12432.43 (approx.)
  • C’s share = (15/37) * 46000 = Rs. 18648.65 (approx.)

Example 4: Reverse Proportion (Trick 3)

If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?

• Solution:
  • This is an inverse proportion problem (more workers, less time).
  • Let the number of workers required be ‘x’.
  • 15 * 48 = x * 30
  • x = (15 * 48) / 30 = 24
  • Therefore, 24 workers will be required.

5. Practice Problems (Graded Difficulty)

[Easy]

1. The ratio of two numbers is 3:5. If the larger number is 75, find the smaller number.
2. Divide Rs. 640 among A, B, and C in the ratio 2:5:9.

[Medium]

3. Two numbers are in the ratio 5:7. If 9 is added to each, the ratio becomes 7:9. Find the numbers.
4. If A:B = 2:3 and B:C = 4:5, find A:B:C.
5. What number must be added to each of the numbers 7, 16, 43 and 79 so that the sums are in proportion?

[Hard]

6. A bag contains Rs. 480 in the form of one rupee, 50 paise and 25 paise coins. The number of coins are in the ratio 5:6:8. Find the number of coins of each type.
7. A and B invest in a business in the ratio 3:2. If 5% of the total profit goes to charity and A’s share is Rs. 855, find the total profit.

6. Advanced/Case-Based Question

Three partners, A, B, and C, invest Rs. 12,000, Rs. 15,000, and Rs. 18,000 respectively in a business. After 4 months, A withdraws Rs. 2,000. After 6 months from the start of the business, B adds Rs. 3,000. C continues with his initial investment. At the end of the year, the total profit is Rs. 36,000. Determine the share of each partner in the profit. Show your calculations and explain the reasoning behind each step.

This guide provides a comprehensive framework for understanding and solving ratio, proportion, and partnership problems. Practice consistently, and you’ll be well-prepared for any quantitative aptitude test. Good luck!