Ratio & Proportion

Master the concepts of ratios and proportions with our comprehensive guide. Learn about direct and inverse proportions, applications, and problem-solving techniques.

Table of Contents

Simplified Quantitative Formulas: Ratio and Proportion

  • Ratio: Compares two similar quantities by division. E.g., 2 apples to 3 bananas is 2:3 (or 2/3).
  • Same Units: Only form a ratio of quantities with the same units or nature.
  • Order Matters: The order in a ratio is important. 2:3 ≠ 3:2.
  • Simplifying Ratios: Multiplying or dividing both parts of a ratio by the same non-zero number doesn't change the ratio. E.g., 2:3 = 4:6 = 1:1.5.
  • Comparing Ratios: Treat ratios like fractions. To compare a:b and p:q, cross-multiply: if aq > bp, then a:b > p:q.
  • Types of Ratios: Greater (antecedent > consequent, e.g., 5:3), Lesser (antecedent < consequent, e.g., 3:5), or Equal (1:1).
  • Effects of Addition/Subtraction: Adding the same positive number to both parts of a ratio will decrease the ratio if it was >1, and increase it if it was <1. Removing does the opposite.
  • Proportion: A proportion states two ratios are equal. E.g., 2:3 = 8:12.
  • Cross-Multiplication: In a proportion a:b = c:d, the products of extremes and means are equal: ad = bc.
  • Continued Proportion: If a, b, c are in continued proportion, a:b = b:c. Here b is the mean proportional, and b² = a×c.
  • Third & Fourth Proportionals: If a:b = b:c, then c is the third proportional to a and b. If a:b = c:d, then d is the fourth proportional to a, b, c.
  • Variations: Direct (x ∝ y), Inverse (x ∝ 1/y), Chain (x ∝ y, y ∝ z ⇒ x ∝ z), and sum (x ∝ y and x ∝ z ⇒ x ∝ (y+z)).
  • Equal Ratios Trick: If a/b = c/d = e/f = ..., then (a+c+e+...)/(b+d+f+...) = common ratio.

What do these mean? (Super Simple Explanations & Examples)

  • Ratio: If you have 2 apples and 3 bananas, the ratio of apples to bananas is 2:3.
    Example: 4 apples and 6 bananas = 4:6 = 2:3.
  • Same Units: You can only compare things that are alike. 5 meters to 5 kilograms is not a valid ratio.
  • Order Matters: 2:3 is not the same as 3:2. The first number always matches the first thing you mention.
  • Simplifying Ratios: 4 apples and 6 bananas is the same as 2:3, because both numbers can be divided by 2.
  • Comparing Ratios: To compare 2:3 and 3:4, cross-multiply: 2×4 = 8, 3×3 = 9. Since 8 < 9, 2:3 < 3:4.
  • Types of Ratios: 5:3 is a greater ratio, 3:5 is a lesser ratio, 5:5 = 1:1 is equality.
  • Effects of Addition/Subtraction: If you add the same number to both sides, a >1 ratio decreases, a <1 ratio increases.
  • Proportion: 2:3 = 8:12 because both are 2/3.
  • Cross-Multiplication: If a:b = c:d, then a×d = b×c. E.g., 2:3 = 8:12 because 2×12 = 3×8 = 24.
  • Continued Proportion: If a:b = b:c, then b is the mean proportional. E.g., 2:4 = 4:8, so 4 is mean proportional between 2 and 8.
  • Third & Fourth Proportionals: If a:b = b:c, c is third proportional. If a:b = c:d, d is fourth proportional.
  • Variations: If x ∝ y, then x = k*y. If x ∝ 1/y, then x = k*(1/y).
    Example: If 2 workers take 4 hours, 4 workers take 2 hours (inverse proportion).
  • Equal Ratios Trick: If a/b = c/d = e/f, then (a+c+e)/(b+d+f) = common ratio.
    Example: 2/3 = 4/6 = 6/9, so (2+4+6)/(3+6+9) = 12/18 = 2/3.

1. Basic Concepts

(a) Understanding Ratios

A ratio is a comparison of two quantities of the same kind using division.

Basic Ratio Formula:

Ratio of a to b = a:b = a/b

Properties:

  • Ratios can be simplified
  • Ratios can be multiplied or divided by the same number
  • Ratios can be added or subtracted when they have the same denominator

Example 1: Basic Ratio

If there are 20 boys and 30 girls in a class:

Ratio of boys to girls = 20:30 = 2:3

Ratio of girls to boys = 30:20 = 3:2

Ratio Visualization

Boys: 40%

Girls: 60%

(b) Types of Ratios

Different types of ratios and their applications.

Common Ratio Types:

  • Simple Ratio: a:b
  • Compound Ratio: (a:b) × (c:d) = (ac:bd)
  • Duplicate Ratio: (a:b)² = (a²:b²)
  • Triplicate Ratio: (a:b)³ = (a³:b³)
  • Sub-duplicate Ratio: √(a:b) = (√a:√b)
  • Sub-triplicate Ratio: ∛(a:b) = (∛a:∛b)

Example 2: Compound Ratio

If A:B = 2:3 and B:C = 4:5

Then A:B:C = 2:3:5

Compound ratio = (2:3) × (4:5) = 8:15

2. Proportions

(a) Direct Proportion

When two quantities increase or decrease together in the same ratio.

Direct Proportion Formula:

If a ∝ b, then a = kb

where k is the constant of proportionality

Example 3: Direct Proportion

If 5 books cost ₹250, then 8 books will cost:

Cost ∝ Number of books

Cost = k × Number of books

250 = k × 5

k = 50

Cost for 8 books = 50 × 8 = ₹400

(b) Inverse Proportion

When one quantity increases while the other decreases in the same ratio.

Inverse Proportion Formula:

If a ∝ 1/b, then a = k/b

where k is the constant of proportionality

Example 4: Inverse Proportion

If 10 workers can complete a work in 6 days, then 15 workers will take:

Number of workers ∝ 1/Time taken

10 × 6 = 15 × x

x = 4 days

3. Applications

(a) Partnership

Distribution of profits based on investment ratios.

Partnership Formula:

Profit Share ∝ Investment × Time

Profit Share = (Investment × Time) / Total (Investment × Time)

Example 5: Partnership

A invests ₹5000 for 6 months

B invests ₹6000 for 4 months

Total profit = ₹1000

Profit ratio = (5000 × 6):(6000 × 4) = 5:4

A's share = (5/9) × 1000 = ₹555.56

B's share = (4/9) × 1000 = ₹444.44

(b) Time and Work

Application of ratios in work efficiency problems.

Work Efficiency Formula:

Work Done ∝ Time × Efficiency

Efficiency Ratio = Time taken by B/Time taken by A

Example 6: Work Efficiency

A takes 6 days, B takes 8 days

Efficiency ratio = 8:6 = 4:3

A is 4/3 times more efficient than B

4. Mixtures & Alligations

(a) Alligation Rule

Method to find the ratio in which two or more ingredients must be mixed.

Alligation Formula:

Quantity of cheaper/Quantity of dearer = (Price of dearer - Mean price)/(Mean price - Price of cheaper)

Example 7: Alligation

Mix two types of rice costing ₹20/kg and ₹30/kg to get rice at ₹25/kg

Ratio = (30-25):(25-20) = 5:5 = 1:1

Mix equal quantities of both types

(b) Mixture Problems

Solving problems involving mixtures of different concentrations.

Mixture Formula:

Final concentration = (C₁V₁ + C₂V₂)/(V₁ + V₂)

where C is concentration and V is volume

Example 8: Mixture

Mix 2L of 20% solution with 3L of 30% solution

Final concentration = (20×2 + 30×3)/(2+3) = 26%

5. Advanced Concepts

(a) Continued Proportion

When three or more quantities are in proportion.

Continued Proportion Formula:

If a:b = b:c, then b² = ac

b is called the mean proportional

Example 9: Continued Proportion

Find the mean proportional between 4 and 9

Let the mean proportional be x

4:x = x:9

x² = 4 × 9 = 36

x = 6

(b) Variation

Different types of variation and their applications.

Variation Formulas:

  • Direct Variation: y = kx
  • Inverse Variation: y = k/x
  • Joint Variation: z = kxy
  • Combined Variation: z = kx/y

Example 10: Combined Variation

z varies directly as x and inversely as y

When x = 4, y = 2, z = 6

Find z when x = 6, y = 3

z = kx/y

6 = k(4/2)

k = 3

z = 3(6/3) = 6

(c) Important Theorems

Key Theorems:

  1. If a:b = c:d, then a+b:a-b = c+d:c-d (Componendo-Dividendo)
  2. If a:b = c:d, then a+c:b+d = a:b = c:d
  3. If a:b = c:d, then a×c:b×d = a²:b²
  4. If a:b = c:d, then (a+b)²:(a-b)² = (c+d)²:(c-d)²

Example 11: Componendo-Dividendo

If a:b = 3:2, find (a+b):(a-b)

Using Componendo-Dividendo:

(a+b):(a-b) = (3+2):(3-2) = 5:1

Practice Questions

Test your understanding of Ratio & Proportion with 20 fully solved, step-by-step questions designed for beginners.

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