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Understanding Salary, Deductions, and Sequences - VOCABULARY Flashcards

    Understanding Income and Compensation

  • Course context: Don Bosco Mandaluyong, General Mathematics TSY 2025-2026

  • Objectives (salary-related):

    • Convert annual salary to monthly and weekly accurately

    • Use the formulas correctly in varied real-world scenarios

    • Demonstrate integrity by valuing honest and transparent salary computations

  • Quick recall prompts (Brain Teaser):

    • If a government employee earns ₱420,000 a year, how much per month? (use monthly formula)

    • If given an annual salary, how can you verify fair weekly pay? (conceptual integrity)

Key Formulas for Salary Calculations

  • Monthly Salary from Annual Salary:
    \text{Monthly Salary} = \frac{\text{Annual Salary}}{12}

  • Weekly Salary from Annual Salary:
    \text{Weekly Salary} = \frac{\text{Annual Salary}}{52}

  • Weekly Salary from Daily Wage (assumes 6 workdays per week):
    \text{Weekly Salary} = \text{Daily Wage} \times 6

  • Monthly Salary from Daily Wage (assuming 24 workdays per month):
    \text{Monthly Salary} = \text{Daily Wage} \times 24

  • Overtime pay (example rule): overtime = 1.25 × regular hourly rate

  • Regular hourly rate from annual salary (for 40-hour week example):
    \text{Regular Hourly Rate} = \frac{\text{Annual Salary}}{52 \times 40}

Salary Breakdown Challenge (given by the activity)

  • Anna: Annual ₱360,000

    • Monthly = ₱360,000 ÷ 12 = ₱30,000

    • Weekly = ₱360,000 ÷ 52 ≈ ₱6,923.08

  • Bryan: Annual ₱420,000

    • Monthly = ₱420,000 ÷ 12 = ₱35,000

    • Weekly ≈ ₱420,000 ÷ 52 ≈ ₱8,076.92

  • Carla: Annual ₱180,000

    • Monthly = ₱180,000 ÷ 12 = ₱15,000

    • Weekly ≈ ₱180,000 ÷ 52 ≈ ₱3,461.54

  • Guiding questions:

    • Who earns the most per week? → Bryan

    • Why know weekly/monthly from annual offers? → Facilitates budgeting, comparisons, and financial planning; ensures fair compensation when negotiating offers

Practical Problems: Weekly and Monthly Salary Calculations

  • Problem 1: Lani ( Cebu City nurse )

    • Annual salary: ₱420,000; weekly hours: 40; overtime this week: 5 hours; overtime rate = 1.25 × regular hourly rate

    • Regular hourly rate: \text{RHR} = \frac{420{,}000}{52 \times 40} \approx ₱201.92/h

    • Overtime rate: ₱201.92 \times 1.25 \approx ₱252.40/h

    • Regular weekly pay: 40 \times ₱201.92 \approx ₱8{,}076.92

    • Overtime pay: 5 \times ₱252.40 ≈ ₱1{,}262.02

    • Total weekly pay (this week): ₱8{,}076.92 + ₱1{,}262.02 ≈ ₱9{,}338.94

  • Problem 2: Joseph (Car sales agent)

    • Annual base: ₱240,000

    • Commission: 2% of sales; this week sales = ₱600,000 → Commission = 0.02 × 600,000 = ₱12,000

    • Weekly base pay: \frac{240{,}000}{52} ≈ ₱4{,}615.38

    • Total weekly income: ₱4{,}615.38 + ₱12,000 = ₱16{,}615.38

  • Problem 3: Mr. Declaro (teacher)

    • Annual salary: ₱384,000 → Monthly = ₱32,000

    • Savings goal: 30% of monthly salary

    • Savings per month: 0.30 × ₱32,000 = ₱9,600

  • Problem 4: Alyssa (freelance insurance agent)

    • Commission: 10% of ₱500,000 = ₱50,000

    • Monthly allowance: ₱5,000

    • Total monthly earnings: ₱50,000 + ₱5,000 = ₱55,000

Additional Individual Work (Weekly and Monthly Earnings Variants)

  • Paolo: annual ₱324,000; paid weekly

    • Weekly earnings: ₱324,000 ÷ 52 = ₱6,230.77

  • Mae (digital embroidery, home-based):

    • Pay per item: ₱60

    • Items per week: 50 → item pay: 50 × ₱60 = ₱3,000

    • Internet allowance: ₱1,200 per month → weekly portion ≈ ₱300

    • Total weekly earnings: ₱3,000 + ₱300 = ₱3,300

  • Rodel (home-based seamer):

    • Pay per mask: ₱35; masks per day: 80; days per month: 25

    • Monthly income: 80 × 35 × 25 = ₱70,000

    • Savings plan: 40% of income → ₱28,000 per month

Value and Ethics in Salary Calculations

  • Value Spotlight: integrity protects both worker and employer; transparency in computations builds trust.

  • Question prompts emphasize ethical handling of earnings and payroll data

Applications of Percentage in Financial Contexts

  • Objectives: Apply percentage changes, compute mark-ups, discounts, VAT, and profit/loss; demonstrate integrity in handling financial data

Percent Challenge: Budget and Pricing Scenarios

  • Group 1 (Budget): Starting monthly groceries ₱1,000; price up by 10% → new cost = ₱1,000 × (1 + 0.10) = ₱1,100; extra funds needed = ₱100

  • Group 2 (Discount): Shirt ₱800 now on sale at 25% discount → final price = ₱800 × (1 − 0.25) = ₱600

  • Group 3 (VAT): Appliance at ₱15,000; add 12% VAT → final price = ₱15{,}000 × 1.12 = ₱16{,}800

  • Group 4 (Profit): Bag bought ₱450, sold ₱600 → profit = ₱150; profit percentage = (₱150 ÷ ₱450) × 100% ≈ 33.33%

Daily Life Computations (Practice Tables)

  • Scenario analyses include % increases, decreases, and VAT applications (examples):

    • Gasoline price: ₱55 → ₱66; % increase = (66−55)/55 = 11/55 = 20%

    • Shirt: ₱500 with 15% discount → final price ₱500 × (1 − 0.15) = ₱425; amount of decrease = ₱75

    • Product cost: ₱150 → sold ₱210; profit = ₱60; profit % = 60/150 × 100% = 40%

    • VAT on ₱1,000 (12%): VAT amount = ₱1,000 × 0.12 = ₱120; VAT-inclusive price = ₱1,120 when base price is ₱1,000

Percentage Change and Inflation (Concepts)

  • Percentage Change = ((New value − Original value) / Original value) × 100%

    • Positive percentage indicates an increase; negative indicates a decrease

  • Inflation and Effects

    • Inflation: general rise in prices → decreases purchasing power

    • Wage adjustment: wages should keep pace with inflation to maintain living standards

    • Rate of inflation is a key parameter for wage policy and cost-of-living assessments

Mark-Ups, Discounts, and VAT (Business context)

  • Cost Price (CP) vs Selling Price (SP)

    • Mark-up percentage = ((SP − CP) / CP) × 100%

  • Discounts

    • Discount Price = Original Price × (1 − Discount Rate)

  • Value Added Tax (VAT)

    • VAT rate in the Philippines: 12%

    • VAT-Inclusive price to VAT-Exclusive (base price) conversion:

    • VAT-Exclusive Price = VAT-Inclusive Price ÷ 1.12

    • VAT Amount = VAT-Inclusive Price − VAT-Exclusive Price

Examples and Practice from the Module

  • Example 1: Mariel purchases school items with discounts; compute total discount and final amount

  • Example 2: VAT-inclusive price example; derive VAT-exclusive price and VAT amount

  • Example 3: Inflation comparisons between 2020 and 2024 for lunch and allowances

  • Example 4: Compare VAT and mark-up in various scenarios; decision-making under price changes

Arithmetic and Geometric Sequences (Foundational Concepts)

  • Arithmetic Sequence

    • Definition: a sequence where the difference between consecutive terms is constant (common difference) d

    • First term: a1

    • nth term: an = a1 + (n - 1)d

    • Examples:

    • 2, 4, 6, 8, … has d = 2

    • Find the 8th term: a8 = a1 + 7d

  • Geometric Sequence

    • Definition: each term is obtained by multiplying the previous term by a fixed nonzero number r (common ratio)

    • First term: a1

    • nth term: an = a1 r^{n-1}

    • Examples:

    • 4, 12, 36, … (r = 3)

  • Geometric Series

    • Finite geometric series: sum of first n terms

    • Sn = a1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)

    • Infinite geometric series: converges if |r| < 1

    • S\infty = \frac{a1}{1 - r}

  • Applications and Examples

    • Solve for nth terms, sums, and partial sums

    • Real-life connections: population growth, savings with compound growth, etc.

Sigma Notation and Series (Summation Conventions)

  • Sigma notation (Σ) basics

    • Used to express sums of sequences: a1 + a2 + a3 + … + an

    • Example conversion: \sum{k=1}^{n} ak

  • Common forms

    • Arithmetic series: Sn = \frac{n}{2} (a1 + an) or equivalently Sn = \frac{n}{2} [2a_1 + (n-1)d]

    • Geometric series (finite): Sn = a1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)

    • Geometric series (infinite): S\infty = \frac{a1}{1 - r} \quad (|r| < 1)

  • Sigma notation practice

    • Rewrite series as sums, expand from sigma notation, evaluate partial sums

    • Examples include: 1 + 2 + 3 + 4, 3 + 9 + 27 + 81 + …, etc.

Arithmetic and Geometric Series – Worked Contexts

  • Rewriting and evaluating finite/infinite series in sigma notation

  • Important cautions

    • Distinguish between a geometric sequence and a geometric series (the latter is a sum of a geometric sequence)

    • Correctly apply formulas for finite vs infinite sums

Investments and Savings Applications

  • Compound interest basics

    • Future value with annual compounding: A = P (1 + r)^t

    • P = principal, r = annual rate, t = years

  • Example problem set (from the module)

    • Bank deposits with 5% annual interest, compounded annually

    • For a deposit of ₱20,000 over 10 years: A = 20{,}000 (1 + 0.05)^{10}

    • Deposit of ₱15,000 at 6% for 8 years: A = 15{,}000 (1 + 0.06)^8

    • Wedding fund with 5.5% annual interest, compounded annually, ₱40,000 for 9 years: A = 40{,}000 (1 + 0.055)^9

  • Conceptual takeaways

    • Regularly saving with compounding yields exponential growth

    • Compare different rates and compounding periods to plan long-term finances

Patterns in Nature and Art

  • Patterns and sequences are used to observe growth and repetition in nature and artifacts

  • Exploration prompts include video-guided observations and questions about growth, patterns, and next terms

Graphical Analysis and Pattern Recognition in Enrollment Data

  • Observations: analyze trends over school years; determine if changes are additive (arithmetic) or multiplicative (geometric)

  • Predict future enrollment: use identified pattern to estimate 2025–2026 and beyond

Think-Pair-Share and Critical Thinking Prompts

  • Engage with problems about sequences, series, and practical payroll scenarios

  • Emphasize reasoning and justifications in solutions

Quick Reference: Key Terms and Formulas

  • Gross Income: total earnings before deductions

    • Gross Income = Salary + Allowances + Overtime + Commission + Piecework (as applicable)

  • Net Pay (Take-Home Pay):
    \text{Net Pay} = \text{Gross Income} - \text{Total Deductions}

  • Deductions (typical in the Philippines):

    • SSS (Social Security System) – private sector employees; purpose: retirement, disability, etc.

    • GSIS – government employees; rate around 9% (employee share)

    • Pag-IBIG (HDMF) – housing savings; employee rate often 2% (with caps)

    • PhilHealth – healthcare; rate around 5% (shared between employee and employer)

    • Withholding Tax – income tax; rate based on tax brackets

    • Other deductions – loans, cash advances, insurance, union dues, absences

  • Important relationships:

    • Net Pay depends on gross income and total deductions

    • Deductions depend on government-m mandated schemes and employee agreements

Note on Payslips and Ethics

  • Payslip components help employees understand how deductions are calculated

  • Understanding net income supports budgeting and financial planning

  • Ethical practice: accurate reporting of gross, deductions, and net pay; avoid misreporting or overstatement of taxes or VAT

Practice Prompt Answers (Highlights)

  • Percentage change: If price increases from Pold to Pnew, % change = ((Pnew − Pold)/P_old) × 100%

  • VAT basics: VAT rate = 12% in this context; VAT-inclusive vs VAT-exclusive pricing

    • VAT-inclusive price = base price × (1 + 0.12)

    • VAT-exclusive price = VAT-inclusive price ÷ 1.12

    • VAT amount = VAT-inclusive − VAT-exclusive

  • Example: If price before VAT is ₱1,000, VAT = ₱120; final (VAT-inclusive) = ₱1,120

Note on Transcripts and Worksheets

  • The content combines salary calculations, percentage, percent-changes, VAT, and sequence theory

  • A number of activities involve converting annual salaries to monthly/weekly, computing deductions, and solving problems using arithmetic and geometric series

  • Kuta Software (Infinite Algebra 2) worksheets included as supplementary practice on sigma notation and series (rewriting, partial sums, and sigma-based problems)