Percentage Increases and Decreases Prices and Wages

Financial Mathematics: Percentages, Prices, and Markup

• Percentage Increase and Decrease

  • Percentage Increase: Shows how much a value has increased compared to its original amount. It is commonly applied in everyday situations such as salary increases, price changes, tuition fees, utility bills, and business profits.
  • Percentage Decrease: Shows how much a value has decreased compared to its original amount. It is commonly applied when there are discounts on products, salary deductions, price reductions, asset depreciation, or decreases in expenses.
  • General Formula for New Value:$\text{New Value} = \text{Original Value} + (\text{Original Value} \times \text{Percentage Rate})$

• Markup and Selling Price

  • Markup: The amount added to the original cost of an item to arrive at its final selling price.
  • Selling Price Formula:$\text{Selling Price} = \text{Cost} + \text{Markup}$
  • Markup Formula:$\text{Markup} = \text{Selling Price} - \text{Cost}$
  • Markup Rate Formula:$\text{Markup Rate} = \frac{\text{Markup}}{\text{Cost}} \times 100\%$
  • Selling Price with Markup Rate:$\text{Selling Price} = \text{Cost} + (\text{Cost} \times \text{Markup Rate})$

• List Price, Discounts, and Net Price

  • List Price: The original price of a product or service before any discount is applied.
  • Discount Rate: The percentage decrease applied to the list price.
  • Discount Price (Discount Amount): The actual monetary reduction calculated from the list price.         $\text{Discount} = \text{List Price} \times \text{Discount Rate}$
  • Net Price: The final amount a customer pays after the discount has been subtracted from the list price.         $\text{Net Price} = \text{List Price} - \text{Discount}$

• Value-Added Tax (VAT)

  • VAT Rate: The Value-Added Tax rate on the sale of services and use or lease of properties is $12\%$ of gross receipts derived from the sales or exchange of services, including the use or lease of properties.
  • VAT Formulas:
    • $V = \text{VAT Amount}$
    • $V_e = \text{VAT-Exclusive Price}$
    • $V_i = \text{VAT-Inclusive Price}$
    • Calculation of VAT amount:             $V = (0.12)V_e$
    • Calculation of VAT-inclusive price:             $V_i = V_e + V$$V_i = (1.12)V_e$
    • Calculation of VAT-exclusive price from VAT-inclusive price:             $V_e = \frac{V_i}{1.12}$

• Profit and Loss Calculations

  • Profit: Occurs when the selling price of a product is greater than its cost price.         $\text{Profit} = \text{Selling Price} - \text{Cost Price}$
  • Loss: Occurs when the selling price of a product is less than its cost price.         $\text{Loss} = \text{Cost Price} - \text{Selling Price}$
  • Profit Percentage:$\text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price}} \times 100$
  • Loss Percentage:$\text{Loss Percentage} = \frac{\text{Loss}}{\text{Cost Price}} \times 100$

Earnings, Compensation, and Deductions

• Salaries vs. Wages

  • Salary: A form of earnings or compensation received by employees in exchange for services rendered that is typically received on a monthly or bi-monthly fixed or regular basis.
  • Wage: A form of earnings or compensation received by employees in exchange for services rendered that typically refers to payments based on the number of hours worked per day.
  • Standard Working Time Conversion:$40\,\text{hours/week} \times 52\,\text{weeks/year} = 2{,}080\,\text{hours/year}$
  • Pay Period Frequency Rates:
    • Monthly: $12$ pay periods per year
    • Bi-monthly: $24$ pay periods per year
    • Weekly: $52$ pay periods per year
    • Total annual work hours: $2{,}080$ hours

• Overtime Pay Structure

  • Overtime Pay: Additional compensation intended to reward employees for extra work performed beyond regular hours. To determine total earnings, overtime pay is added to the regular salary.
  • Variables:
    • $DR$ = Daily basic rate
    • $HR$ = Hourly rate
    • $OT$ = Overtime hours
  • Overtime Rate Multipliers:
    • Regular Workday Overtime Rate: $125\% \times HR$
    • Rest Day or Special Holiday Overtime Rate: $130\% \times HR$
    • Special Holiday Falling on Rest Day Overtime Rate: $150\% \times HR$
    • Regular Holiday Overtime Rate: $200\% \times HR$
    • Regular Holiday Falling on Rest Day Overtime Rate: $260\% \times HR$

• Commissions and Graduated Rates

  • Commission: Amount of money earned based on the value of products or services sold. Higher sales result in higher commissions earned.         $\text{Commission} = \text{Sales} \times \text{Commission Rate}$
  • Salary Plus Commission: Total compensation when combining a basic salary with commission earnings.         $\text{Total Earnings} = \text{Basic Salary} + \text{Commission}$
    • Example: If an employee has a basic salary of $\text{₱}12{,}000$ and earns a commission of $\text{₱}4{,}800$, total compensation is:             $\text{Total Earnings} = \text{₱}12{,}000 + \text{₱}4{,}800 = \text{₱}16{,}800$
  • Graduated Commission: A system where the commission rate increases as the sales volume reaches higher tier thresholds.
    • Example Scenario (Ann's Insurance Sales):             Ann earns a graduated commission with rates structure:
      • $5\%$ on the first $\text{₱}5{,}000$ worth of sales
      • $10\%$ on the next $\text{₱}15{,}000$ worth of sales
      • $15\%$ on sales in excess of $\text{₱}20{,}000$
      • Calculation for total sales of $\text{₱}25{,}000$:
        • First $\text{₱}5{,}000$ sales tier: $\text{₱}5{,}000 \times 0.05 = \text{₱}250$
        • Next $\text{₱}15{,}000$ sales tier: $\text{₱}15{,}000 \times 0.10 = \text{₱}1{,}500$
        • Remaining $\text{₱}5{,}000$ sales tier (excess over $\text{₱}20{,}000$): $\text{₱}5{,}000 \times 0.15 = \text{₱}750$
        • Total Commission: $\text{₱}250 + \text{₱}1{,}500 + \text{₱}750 = \text{₱}2{,}500$

• Gross Income, Deductions, and Net Income

  • Gross Income: Total earnings before any deductions are subtracted.
  • Net Income: The final amount received after all required and voluntary deductions are subtracted.
  • Benefits: Additional forms of compensation that improve an employee's financial security, health, and overall well-being. Provided either as direct cash or as non-cash services/privileges.
  • Net Income Formula:$\text{Net Income} = \text{Gross Income} - \text{Total Deductions}$
  • Comprehensive Computation Example:
    • Income Items:
      • Monthly Salary: $\text{₱}30{,}000$
      • Overtime Pay: $\text{₱}1{,}250$
      • Rice Allowance: $\text{₱}1{,}000$
      • Transportation Allowance: $\text{₱}1{,}500$
      • $\text{Gross Income} = \text{₱}30{,}000 + \text{₱}1{,}250 + \text{₱}1{,}000 + \text{₱}1{,}500 = \text{₱}33{,}750$
    • Deduction Items:
      • Tax: $\text{₱}2{,}500$
      • SSS: $\text{₱}1{,}000$
      • PhilHealth: $\text{₱}800$
      • Pag-IBIG: $\text{₱}200$
      • $\text{Total Deductions} = \text{₱}2{,}500 + \text{₱}1{,}000 + \text{₱}800 + \text{₱}200 = \text{₱}4{,}500$
    • Final Net Income:$\text{Net Income} = \text{₱}33{,}750 - \text{₱}4{,}500 = \text{₱}29{,}250$

Patterns in Mathematics, Art, and Nature

• Understanding Patterns

  • Pattern: A repeated or predictable arrangement of numbers, shapes, colors, letters, sounds, movements, or objects. Every pattern follows a governing rule that dictates how it is formed.

• Patterns in Art and Architecture

  • Patterns in art are repeated designs, shapes, colors, lines, or symbols arranged in an organized way. Artists and architects use patterns to make their creations visually appealing, balanced, and meaningful.

• Patterns in Nature
  • Unlike human-made patterns, natural patterns do not always repeat perfectly. Instead, they grow, branch, spiral, or transform according to natural biological and physical laws.

• The Fibonacci Sequence and Golden Ratio

  • Fibonacci Sequence: A special numerical pattern where each number is the exact sum of the two preceding numbers.
    • Sequence sequence terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$
  • Golden Ratio (\phi): When two quantities are in the Golden Ratio, the ratio of the longer part to the shorter part is approximately $1.618$. It is associated with visual harmony, balance, and aesthetics in nature and design.
  • Ratio Convergence of Fibonacci Consecutive Terms:
    • $5 \div 3 = 1.667$
    • $8 \div 5 = 1.600$
    • $13 \div 8 = 1.625$
    • $21 \div 13 = 1.615$
    • $34 \div 21 = 1.619$
    • $55 \div 34 = 1.618$

• Binet's Formula for Fibonacci Numbers

  • Binet's Formula: An explicit formula used to calculate the $n\text{th}$ term of the Fibonacci sequence directly without sequential addition.         $F_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$
    • Where:
      • $F_n$ = $n\text{th}$ Fibonacci number
      • $\phi$ = Golden ratio $\approx 1.618034$
  • Worked Application Examples:
    • Example 1: Find $F_{22}$$F_{22} = \frac{(1.618034)^{22} - (1 - 1.618034)^{22}}{\sqrt{5}}$$F_{22} = 17{,}711$
    • Example 2: Find $F_{15}$$F_{15} = \frac{(1.618034)^{15} - (1 - 1.618034)^{15}}{\sqrt{5}}$$F_{15} = 610$
    • Example 3: Find $F_{30}$$F_{30} = \frac{(1.618034)^{30} - (1 - 1.618034)^{30}}{\sqrt{5}}$$F_{30} = 832{,}040$

Mathematical Sequences and Series

• Definitions and Terminology

  • Sequence: An ordered list of numbers that follows a specific pattern or underlying rule.
    • Example Sequence: $2, 4, 6, 8, 10$
    • $1\text{st term} (a_1) = 2$
    • $2\text{nd term} (a_2) = 4$
    • $3\text{rd term} (a_3) = 6$
    • $4\text{th term} (a_4) = 8$
    • $5\text{th term} (a_5) = 10$
    • Rule: Each term increases by $2$.
  • Series: The sum of the terms of a sequence.
    • Example Series: $2 + 4 + 6 + 8 + 10 = 30$

• Mathematical Notation for Series

  • For a sequence $a_1, a_2, a_3, \dots, a_n$, the corresponding series $S_n$ is:         $S_n = a_1 + a_2 + a_3 + \dots + a_n$
    • Where:
      • $S_n$ = sum of the first $n$ terms ($n\text{th}$ partial sum)
      • $a_1$ = first term
      • $a_2, a_3, \dots, a_n$ = succeeding terms
      • $n$ = total number of terms
  • Sigma Notation (\sum):$S_n = \sum_{i=1}^{n} a_i$
    • $\sum$ symbol represents summation ("add").
    • $i = 1$ specifies starting index (first term).
    • $n$ specifies ending index (last term).
    • $a_i$ represents the explicit rule or term formula of the sequence.

• Worked Examples of Partial Sums and Series Expansion

  • Example 1: Given the sequence $10, 20, 30, 40, 50, 60, 70, 80, 90, 100, \dots$
    • Find $S_5$:             $S_5 = a_1 + a_2 + a_3 + a_4 + a_5$$S_5 = 10 + 20 + 30 + 40 + 50 = 150$
    • Find $S_7$:             $S_7 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7$$S_7 = 10 + 20 + 30 + 40 + 50 + 60 + 70 = 280$
  • Example 2: Evaluate the explicit series $S_4 = \sum_{i=1}^{4} (7i + 2)$
    • $i = 1$ term: $7(1) + 2 = 9$
    • $i = 2$ term: $7(2) + 2 = 16$
    • $i = 3$ term: $7(3) + 2 = 23$
    • $i = 4$ term: $7(4) + 2 = 30$
    • Sum calculation:             $S_4 = 9 + 16 + 23 + 30 = 78$