Ratio & Proportion

Ratio & Proportion 

⭐Ratio

A ratio is a way to compare two quantities by showing the relative sizes of two or more values. It is one of the fundamental concepts in arithmetic and is widely used in various fields such as finance, marketing, and economics.

Basic Concepts:

• Definition:
• A ratio of two numbers a and b is written as a:b or $\frac{a}{b}$. It indicates how many times one number contains the other.

• Types of Ratios:
• Simple Ratio: A straightforward comparison between two quantities. For example, 4:5.
• Compound Ratio: A combination of two or more ratios. For example, combining the ratios 2:3 and 4:5 yields a compound ratio of (24):(35) = 8:15.

• Properties of Ratios:
• Order Matters: The ratio 2:3 is not the same as 3:2.
• Equivalent Ratios: Ratios that represent the same relationship. For example, 2:3 is equivalent to 4:6, since both simplify to the same ratio.

• Simplifying Ratios:
• To simplify a ratio, divide both terms by their greatest common divisor (GCD). For example, the ratio 8:12 simplifies to 2:3 (GCD of 8 and 12 is 4).

• Ratio in Different Forms:

• Fraction: $\frac{a}{b}$

• Colon: a:b

• Decimal: $\frac{a}{b}$​ as a decimal value

Examples:

• Finding Ratio:
• If there are 15 boys and 20 girls in a class, the ratio of boys to girls is 15:20, which simplifies to 3:4.

• Comparing Ratios:
• To compare 3:4 and 6:8, simplify both to their lowest terms. Both simplify to 3:4, indicating they are equivalent.

⭐Sub-Topics in Ratio and Proportion

1. Types of Ratios:

• Duplicate Ratio: The ratio of the squares of two numbers. If the ratio is a:b, the duplicate ratio is $\frac{a^2}{b^2}$ .

• Triplicate Ratio: The ratio of the cubes of two numbers. If the ratio is a:b, the triplicate ratio is $\frac{a^3}{b^3}$ .

• Sub-duplicate Ratio: The ratio of the square roots of two numbers. If the ratio is a:b, the sub-duplicate ratio is $\frac{\sqrt{a}}{\sqrt{b}}$ ​.

• Sub-triplicate Ratio: The ratio of the cube roots of two numbers. If the ratio is a:b, the sub-triplicate ratio is $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ ​.

• Inverse Ratio: The inverse ratio of a:b is $\frac{1}{a}:\frac{1}{b}$ $or$ b:a.

2. Compound Ratio:

• Definition: The ratio obtained by multiplying the corresponding terms of two or more ratios.
• Example:
  • Given two ratios a:b and c:d, the compound ratio is ac:bd.

3. Continued Proportion:

• Definition: When three quantities a, b, and ccc are in continued proportion, 

a:b = b:c.

• Example:
  • If 2:4 = 4:8, then 2,4,8 are in continued proportion.

4. Mean Proportion:

• Definition: If a:b = b:c, then b is called the mean proportion between a and c.
• Calculation:
  • Mean Proportion, $b=\sqrt{ac}$​.

5. Fourth Proportional:

• Definition: If a:b = c:d, then d is called the fourth proportional to a, b, and c.
• Calculation:
  • Fourth Proportional, $d=\frac{bc}{a}$​.

6. Third Proportional:

• Definition: If a:b = b:c, then ccc is called the third proportional to a and b.
• Calculation:
  • Third Proportional, $c=\frac{b^2}{a}$​.

⭐Proportion

A proportion states that two ratios are equal. It is an equation that expresses the equality of two ratios, and it's used to solve problems involving the comparison of quantities.

Basic Concepts:

• Definition:
• Two ratios a:b and c:d are in proportion if $\frac{a}{b}$ = $\frac{c}{d}$​. This is written as a:b::c:d.

• Properties of Proportions:
• Cross Multiplication: If a:b::c:d, then ad=bc.
• Means and Extremes: In a:b::c:d, a and d are the extremes, and b and c are the means.

• Types of Proportions:
• Direct Proportion: Two quantities increase or decrease together in the same ratio. For example, if the number of items bought increases, the total cost increases proportionally.
• Inverse Proportion: An increase in one quantity results in a proportional decrease in the other quantity. For example, if the speed of a vehicle increases, the time taken to cover a fixed distance decreases.

• Proportionality Constant:
  • In direct proportion, the ratio of the two quantities remains constant, i.e., $\frac{a}{b}$ = k (constant).

Examples:

• Solving Proportion Problems:
• If 5 pens cost $10, how much do 8 pens cost?
• Let the cost of 8 pens be x.
• 5:10::8:x ⇒ $\frac{5}{10}=\frac{8}{x}$
• Cross-multiply: 5x = 80 ⇒ x = 16
• So, 8 pens cost $16.

• Direct Proportion:
• If 3 kg of rice costs $15, what is the cost of 5 kg of rice?
• Cost is directly proportional to quantity.
• Let the cost of 5 kg of rice be x.
• $\frac{3}{15}=\frac{5}{x}$​
• Cross-multiply: 3x = 75 ⇒ x = 25
• So, 5 kg of rice costs $25.

• Inverse Proportion:
  • If 4 workers can complete a task in 6 days, how many days will 8 workers take to complete the same task?
    • Workers and days are inversely proportional.
    • Let the number of days 8 workers take be x.
    • 4×6 = $8\times x$
    • 24 = 8x ⇒ x = 3
    • So, 8 workers will take 3 days to complete the task.

Applications in BBA Course:

• Financial Ratios: Understanding ratios like debt-to-equity, profit margins, and liquidity ratios are essential in financial analysis.
• Market Analysis: Ratios are used to analyze market trends, compare performance, and make informed decisions.
• Resource Allocation: Proportions help in budgeting and allocating resources efficiently.

Practice Problems:

• Ratio Problems:
  • Simplify the ratio 45:60.
  • Find the ratio of 120 minutes to 2 hours.
• Proportion Problems:
  • If 7 books cost $21, how much do 10 books cost?
  • If 5 machines can produce 200 units in 8 hours, how many units can 8 machines produce in the same time?

Understanding these fundamental concepts of ratio and proportion is crucial for arithmetic aptitude. These concepts are widely applicable in various aspects of business administration and management, aiding in decision-making, financial analysis, and strategic planning.