Unit 2: Ratio, Proportion, Variation and Arithmetic and Geometric Progressions

1. Percentage

1.1 Definition and Basic Concepts

• Percentage: The term "percent" is derived from the Latin phrase "per centum," which means "by the hundred" or "for every hundred." A percentage is a fraction or a ratio in which the value of a whole is always 100. It is denoted by the symbol %.
• Purpose: Percentages are used to express how large or small one quantity is relative to another quantity. They provide a standardized way to compare different values.

1.2 Formulae and Conversions

Basic Formula:

Percentage = (Part / Whole) × 100

Conversions:

• Percentage to Fraction: Divide by 100 and remove the % symbol.
  • Example: 45% = 45 / 100 = 9 / 20
• Fraction to Percentage: Multiply the fraction by 100 and add the % symbol.
  • Example: 3 / 5 = (3 / 5) × 100% = 60%
• Percentage to Decimal: Divide by 100 (or move the decimal point two places to the left).
  • Example: 75% = 75 / 100 = 0.75
• Decimal to Percentage: Multiply by 100 (or move the decimal point two places to the right).
  • Example: 0.8 = 0.8 × 100% = 80%

1.3 Percentage Change

Percentage change is used to measure the difference in a value over time.

• Percentage Increase: Used when a value grows or increases.
  

  TEXT

      Percentage Increase = [(New Value - Original Value) / Original Value] × 100
      

  • Example: If an employee's salary increases from 45,000, the percentage increase is:
    [(45000 - 40000) / 40000] × 100 = (5000 / 40000) × 100 = 12.5%

• Percentage Decrease: Used when a value shrinks or decreases.
  

  TEXT

      Percentage Decrease = [(Original Value - New Value) / Original Value] × 100
      

  • Example: If a product's price drops from 40, the percentage decrease is:
    [(50 - 40) / 50] × 100 = (10 / 50) × 100 = 20%

1.4 Business Applications

• Profit and Loss:

  • Cost Price (CP): The price at which an item is purchased.
  • Selling Price (SP): The price at which an item is sold.
  • Profit (Gain): If SP > CP, then Profit = SP - CP.
  • Loss: If CP > SP, then Loss = CP - SP.
  • Profit Percentage: (Profit / CP) × 100
  • Loss Percentage: (Loss / CP) × 100
  • Note: Profit and Loss are always calculated on the Cost Price unless otherwise specified.

• Discount:

  • Marked Price (MP) or List Price: The price tagged on an item.
  • Discount: A reduction on the Marked Price.
  • Net Selling Price (SP): SP = MP - Discount
  • Discount Percentage: (Discount / MP) × 100

• Commission and Brokerage:

  • Commission: A fee paid to an agent or employee for transacting a piece of business or performing a service. It is usually calculated as a percentage of the total transaction value.
  • Example: A real estate agent earns a 3% commission on a house sale of $500,000.
    Commission = 3% of 15,000

• Taxes (e.g., GST, VAT):

  • Taxes are often calculated as a percentage of the price of goods or services.
  • Example: If a product costs $200 and the Goods and Services Tax (GST) is 18%, the tax amount is:
    Tax Amount = 18% of 36
Total Price = 36 = $236

2. Ratio

2.1 Definition and Properties

• Ratio: A ratio is a quantitative relation between two amounts showing the number of times one value contains or is contained within the other. It is a comparison of two quantities of the same kind and in the same units.
• Notation: The ratio of a to b is written as a : b or a/b.
  • a is called the antecedent (the first term).
  • b is called the consequent (the second term).
• Properties:
  1. A ratio is a pure number and has no units.
  2. The order of the terms in a ratio is important (i.e., a : b ≠ b : a).
  3. A ratio remains unchanged if both its terms are multiplied or divided by the same non-zero number. E.g., 2 : 4 is the same as 1 : 2.
  4. To compare ratios, it's best to convert them into equivalent fractions with a common denominator or to decimals.

2.2 Types of Ratios

• Inverse Ratio: The inverse ratio of a : b is b : a.
• Compound Ratio: The compound ratio of two or more ratios (a : b, c : d, e : f) is the ratio of the product of their antecedents to the product of their consequents.
  • Compound Ratio = (a × c × e) : (b × d × f)
• Duplicate Ratio: The duplicate ratio of a : b is a² : b².
• Sub-duplicate Ratio: The sub-duplicate ratio of a : b is √a : √b.
• Triplicate Ratio: The triplicate ratio of a : b is a³ : b³.
• Sub-triplicate Ratio: The sub-triplicate ratio of a : b is ³√a : ³√b.

2.3 Application: Dividing a Quantity

To divide a quantity Q into two parts in the ratio a : b:

• Sum of the ratio terms = a + b.
• First Part = Q × [a / (a + b)]
• Second Part = Q × [b / (a + b)]
• Example: Divide $1,500 between two partners in the ratio 2 : 3.
  • Sum of ratios = 2 + 3 = 5.
  • Partner 1's share = 1500 × (2 / 5) = $600.
  • Partner 2's share = 1500 × (3 / 5) = $900.

3. Proportion

3.1 Definition and Properties

• Proportion: A proportion is an equation that states that two ratios are equal. If a : b is equal to c : d, then a, b, c, d are said to be in proportion.
• Notation: a : b :: c : d or a / b = c / d.
• Terms:
  • a and d are called the extremes (outer terms).
  • b and c are called the means (inner terms).
• Fundamental Property: In any proportion, the product of the extremes is equal to the product of the means.
  
  TEXT

      If a : b = c : d, then a × d = b × c
      

• Continued Proportion: Three quantities a, b, c are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third.
  • a : b :: b : c => a / b = b / c => b² = ac.
  • Here, b is called the mean proportional between a and c.

3.2 Properties of Proportion (Rules of Algebra of Proportions)

If a : b = c : d, then the following properties hold true:

1. Invertendo: The inverse of the ratios are also in proportion.
   • b : a = d : c
2. Alternendo: The ratio of the antecedents is equal to the ratio of the consequents.
   • a : c = b : d
3. Componendo: (Adding 1 to both sides of a/b = c/d)
   • (a + b) : b = (c + d) : d
4. Dividendo: (Subtracting 1 from both sides of a/b = c/d)
   • (a - b) : b = (c - d) : d
5. Componendo and Dividendo: A combination of the above two properties.
   • (a + b) : (a - b) = (c + d) : (c - d)

4. Variation

4.1 Direct Variation

• Definition: Two quantities, x and y, are said to be in direct variation if they increase or decrease together in such a way that their ratio remains constant.
• Expression:
  • y varies directly as x.
  • y ∝ x
  • y = kx, where k is the constant of variation.
• Business Example: The total cost of raw materials (y) varies directly with the number of units produced (x). If the cost per unit (k) is $5, then y = 5x.

4.2 Inverse Variation

• Definition: Two quantities, x and y, are said to be in inverse variation if an increase in one quantity causes a proportional decrease in the other, and vice-versa, such that their product remains constant.
• Expression:
  • y varies inversely as x.
  • y ∝ 1/x
  • y = k/x or xy = k, where k is the constant of variation.
• Business Example: The time (y) taken to complete a project varies inversely with the number of workers (x) employed. If more workers are hired, the time to complete the project decreases. xy = k (where k represents total man-hours).

4.3 Joint Variation

• Definition: A quantity is in joint variation with two or more other quantities if it varies directly as their product.
• Expression:
  • z varies jointly as x and y.
  • z ∝ xy
  • z = kxy, where k is the constant of variation.
• Business Example: Simple Interest (I) varies jointly with the Principal (P) and Time (T). The formula is I = RPT, where the rate (R) is the constant of variation.

5. Arithmetic Progression (AP)

5.1 Definition and Terminology

• Arithmetic Progression (AP): A sequence of numbers is called an arithmetic progression if the difference between any term and its preceding term is constant throughout the sequence.
• First Term (a): The first number in the sequence.
• Common Difference (d): The constant difference between consecutive terms. d = T₂ - T₁ = T₃ - T₂, etc.

5.2 General Term and Sum

• The nth Term (Tₙ or aₙ): This formula finds any term in the AP.
  

  TEXT

      Tₙ = a + (n - 1)d
      

  • Tₙ: The value of the nth term.
  • a: The first term.
  • n: The position of the term in the sequence.
  • d: The common difference.
  • Example: Find the 10th term of the AP: 3, 7, 11, 15...
    • a = 3, d = 7 - 3 = 4, n = 10.
    • T₁₀ = 3 + (10 - 1) × 4 = 3 + 9 × 4 = 3 + 36 = 39.

• Sum of the First n Terms (Sₙ): This formula finds the sum of a certain number of terms.

  • Formula 1 (when a, n, d are known):
    
    TEXT

            Sₙ = n/2 [2a + (n - 1)d]
            

  • Formula 2 (when first term a and last term l are known):
    
    TEXT

            Sₙ = n/2 (a + l)
            

  • Example: Find the sum of the first 20 terms of the AP: 2, 5, 8...
    • a = 2, d = 3, n = 20.
    • S₂₀ = 20/2 [2(2) + (20 - 1)3] = 10 [4 + 19 × 3] = 10 [4 + 57] = 10 × 61 = 610.

5.3 Arithmetic Mean (AM)

• The arithmetic mean of two numbers a and b is (a + b) / 2.
• To insert k arithmetic means between two numbers a and b, you create an AP with n = k + 2 terms, where the first term is a and the last term is b. You can then find the common difference d and subsequently the means.

5.4 Business Applications

• Simple Interest: The annual amounts of simple interest form an AP. For a principal P at a rate R, the interest for the 1st, 2nd, 3rd year is PR, PR, PR... The total amount A at the end of each year (P+PR, P+2PR, P+3PR...) forms an AP with common difference PR.
• Straight-Line Depreciation: If an asset worth 800 each year, its value at the end of year 1, 2, 3, etc., will be $9200, $8400, $7600..., which is an AP.
• Salary Increments: An employee joins a company with a salary of 2,500. Their salary in subsequent years will form an AP.

6. Geometric Progression (GP)

6.1 Definition and Terminology

• Geometric Progression (GP): A sequence of non-zero numbers is called a geometric progression if the ratio of any term to its preceding term is constant throughout the sequence.
• First Term (a): The first number in the sequence.
• Common Ratio (r): The constant ratio between consecutive terms. r = T₂ / T₁ = T₃ / T₂, etc.

6.2 General Term and Sum

• The nth Term (Tₙ or aₙ):
  

  TEXT

      Tₙ = ar^(n-1)
      

  • Example: Find the 8th term of the GP: 3, 6, 12, 24...
    • a = 3, r = 6 / 3 = 2, n = 8.
    • T₈ = 3 × 2^(8-1) = 3 × 2⁷ = 3 × 128 = 384.

• Sum of the First n Terms (Sₙ):

  • Formula for r ≠ 1:
    
    TEXT

            Sₙ = a(rⁿ - 1) / (r - 1)   or   Sₙ = a(1 - rⁿ) / (1 - r)
            

    
    (The first is convenient for |r| > 1, the second for |r| < 1).
  • Formula for r = 1: Sₙ = na.
  • Example: Find the sum of the first 5 terms of the GP: 5, 15, 45...
    • a = 5, r = 3, n = 5. Since r > 1:
    • S₅ = 5(3⁵ - 1) / (3 - 1) = 5(243 - 1) / 2 = 5(242) / 2 = 5 × 121 = 605.

• Sum of an Infinite Geometric Progression (S∞):

  • The sum of an infinite GP exists only if the common ratio r is between -1 and 1 (i.e., |r| < 1).
  • Formula:
    
    TEXT

            S∞ = a / (1 - r)
            

  • Example: Find the sum of the infinite GP: 16, 8, 4, 2...
    • a = 16, r = 8 / 16 = 1/2. Since |r| < 1, the sum exists.
    • S∞ = 16 / (1 - 1/2) = 16 / (1/2) = 32.

6.3 Geometric Mean (GM)

• The geometric mean of two positive numbers a and b is √(ab).
• If a, b, c are in GP, then b is the geometric mean of a and c, so b = √(ac).

6.4 Business Applications

• Compound Interest: The total amount in an account with compound interest forms a GP. If principal P is invested at a rate i per period, the amount at the end of period 1, 2, 3... is P(1+i), P(1+i)², P(1+i)³... This is a GP with first term a = P(1+i) and common ratio r = (1+i).
• Reducing-Balance Depreciation: If an asset worth 18000, $16200, $14580... form a GP with r = 0.9.
• Population Growth: If a city's population grows at a constant rate of 2% per year, the population figures for consecutive years form a GP with r = 1.02.
• Annuities: The calculation of the present and future value of annuities involves the sum of terms in a geometric progression.