Fiche de révision : Mastering Percentage Changes and Growth Calculations

📋 Course Outline

1. Percentage Increase Factors
2. Compound Growth Calculations
3. Price Discount Calculations
4. Investment Growth
5. Depreciation of Assets
6. Multiplicative Change
7. Population Growth
8. Tax and Discount Application

📖 1. Percentage Increase Factors

🔑 Key Concepts & Definitions

• Percentage Increase Factor: The multiplier that represents the effect of a percentage increase on a quantity. It is calculated as 1 plus the percentage increase divided by 100. (Author: general mathematical principle)
  Example: A 10% increase corresponds to a percentage increase factor of 1.10.

• Calculation of Change Factor for a Given Percentage Increase: To find the change factor, convert the percentage increase into a decimal and add 1. (Author: basic percentage-to-factor conversion)
  Formula: Change factor = 1 + (percentage increase / 100)

• Applying Percentage Increase Factors Sequentially: When multiple percentage increases are applied one after another, multiply their respective change factors to find the overall effect. (Author: multiplicative property of percentage changes)
  Example: An increase of 10% followed by 20% results in a total change factor of 1.10 × 1.20 = 1.32, equivalent to a 32% increase.

📝 Essential Points

• The percentage increase factor simplifies calculations involving repeated percentage increases by converting them into multiplication operations.
• To determine the new value after a percentage increase, multiply the original value by the change factor.
• When applying multiple percentage increases sequentially, the total change factor is the product of individual change factors (see the example in the last point).
• For example, a 10% increase corresponds to a change factor of 1.10, and a 20% increase corresponds to 1.20.
• The total effect of applying two increases (e.g., 10% then 20%) is found by multiplying their change factors, not by adding the percentages directly.

💡 Key Takeaway

The percentage increase factor provides a straightforward way to calculate the effect of percentage increases, especially when applying multiple increases sequentially, by using multiplication of their respective change factors.

📖 2. Compound Growth Calculations

🔑 Key Concepts & Definitions

• Compound Growth: The process where a quantity increases by a certain percentage over multiple periods, with each increase building upon the previous total. As AUTHOR (date) explains, it involves applying the growth repeatedly, leading to exponential growth rather than linear.

• Calculating Compound Growth Over Multiple Periods: To find the total growth after several periods, multiply the initial amount by the growth factors for each period. For example, if a value grows by a percentage each period, the total after n periods is the initial amount multiplied by the product of all growth factors (see example in the restaurant booking problem).

• Using Growth Factors for Repeated Percentage Increases: The growth factor for a percentage increase of p% is calculated as $1 + \frac{p}{100}$. Repeated increases are modeled by multiplying the initial value by the growth factors for each period, which simplifies calculations of compound growth (see the example of Jorma's pole vault record).

📝 Essential Points

• The growth factor corresponding to a percentage increase of p% is $1 + \frac{p}{100}$. For example, a 10% increase has a growth factor of 1.10, and a 20% increase has a growth factor of 1.20.

• To calculate the total growth over multiple periods, multiply the initial amount by each period's growth factor. For instance, if bookings increase by 10% in the first week and 20% in the second, the total growth factor is $1.10 \times 1.20 = 1.32$, meaning a 32% total increase.

• When dealing with successive percentage increases, the overall growth factor is the product of individual growth factors, which can be used repeatedly for modeling compound growth (see the example of fond value growth over 4 months).

• The concept of compound growth is fundamental in finance, population studies, and any scenario involving repeated percentage changes over time, emphasizing the exponential nature of such growth.

💡 Key Takeaway

Compound growth involves applying a growth factor repeatedly over multiple periods, resulting in exponential increases that are best calculated by multiplying the initial amount by the product of all relevant growth factors.

📖 3. Price Discount Calculations

🔑 Key Concepts & Definitions

Price discount:
A reduction in the original price of a product or service, expressed either as a percentage or a fixed amount, intended to lower the selling price (see section 8 for related tax and discount application).

Calculating successive price discounts:
The process of applying multiple discounts one after another to the current price, rather than summing the percentage discounts and applying them once. This involves multiplying the current price by each discount factor sequentially.

Effect of multiple discounts on original price:
When multiple discounts are applied successively, the final price is lower than if a single combined discount were applied. The overall effect is found by multiplying the individual discount factors, which results in a compounded reduction of the original price.

📝 Essential Points

• A price discount expressed as a percentage (e.g., 12%) corresponds to a discount factor of (1 - 0.12) = 0.88.
• To calculate successive discounts, multiply the current price by each discount factor in sequence. For example, after a 12% discount and an 8% discount, the overall discount factor is 0.88 × 0.92 = 0.8096, meaning the final price is approximately 80.96% of the original.
• The effect of multiple discounts is multiplicative, not additive. Applying two discounts of 10% and 20% results in a total reduction of about 28%, not 30%. This is because each discount reduces the current price, which has already been decreased by the previous discount.

💡 Key Takeaway

Applying multiple discounts successively involves multiplying their respective discount factors, which results in a compounded reduction of the original price, often less than the sum of individual discounts.

📖 4. Investment Growth

🔑 Key Concepts & Definitions

• Investment value growth over time: The increase in the value of an investment as a result of percentage changes applied periodically, reflecting how investments can appreciate or depreciate over multiple periods (see source content for examples of percentage increases and decreases).

• Monthly percentage increase in investment value: The rate at which an investment's value increases each month, expressed as a percentage, which can be used to calculate the new value after each month (e.g., a 3% monthly growth in the fund's value).

• Calculating investment value after several periods: The process of determining the future value of an investment after multiple periods of growth or decline by applying the relevant change factors sequentially, often through multiplication of the individual growth factors for each period.

📝 Essential Points

• Changes in investment value are often expressed as change factors, which are derived from percentage increases or decreases (e.g., a 10% increase corresponds to a change factor of 1.10). These factors are multiplied across periods to find the total growth over time.

• The total growth of an investment after multiple periods is calculated by multiplying the individual change factors for each period, reflecting the compound effect of sequential percentage changes (see example of fund increasing 3% monthly over 4 months).

• When an investment experiences successive percentage increases, the overall change factor is the product of the individual change factors, which can be greater than the sum of the individual percentages due to compounding effects.

• For decreases, the change factor is less than 1 (e.g., a 15% decrease corresponds to 0.85), and multiplying these factors over periods gives the cumulative effect on the investment's value.

💡 Key Takeaway

The growth of an investment over time is best understood through change factors and their multiplication across periods, illustrating how small percentage changes compound to produce significant long-term effects.

📖 5. Depreciation of Assets

🔑 Key Concepts & Definitions

• Asset depreciation (see section 1): The process of allocating the cost of a tangible asset over its useful life, reflecting the reduction in its value due to wear and tear, obsolescence, or age. It is a systematic way to account for the decreasing value of an asset over time.

• Calculating asset value after repeated percentage decreases (see section 4): The process of determining an asset's current value after multiple reductions expressed as percentages. This involves multiplying the initial value by successive change factors (1 minus the percentage decrease divided by 100) for each period.

• Annual percentage decrease in asset value (see section 4): The fixed percentage by which an asset's value diminishes each year. This rate is used to calculate the asset's remaining value after a certain number of years, assuming a consistent rate of depreciation annually.

📝 Essential Points

• Asset depreciation is crucial for financial reporting and tax purposes, enabling businesses to spread the cost of an asset over its useful life (see section 1).
• To find the asset's value after multiple periods of depreciation, multiply the initial value by the change factors for each period, which are derived from the percentage decreases (see section 4).
• The annual percentage decrease provides a straightforward way to estimate how much value an asset loses each year, assuming a constant rate of depreciation.
• For example, if an asset decreases by 15% annually, the remaining value after n years is calculated by multiplying the initial value by (1 - 0.15)^n.
• Understanding the relationship between percentage decreases and change factors helps in planning asset replacement and financial analysis.

💡 Key Takeaway

Asset depreciation involves systematically reducing an asset's value over time using percentage decreases, with the calculation of remaining value after repeated decreases being essential for accurate financial management.

📖 6. Multiplicative Change

🔑 Key Concepts & Definitions

• Multiplicative change as product of change factors: The overall change resulting from multiple sequential changes is found by multiplying their individual change factors. If each change is represented by a change factor, the total change is the product of these factors (see examples 4227, 4234).

• Interpreting multiplication of change factors: When change factors are multiplied, the result indicates the cumulative effect of sequential changes. For example, a 10% increase (factor 1.10) followed by a 20% increase (factor 1.20) results in a total change factor of 1.10 × 1.20 = 1.32, meaning a 32% increase overall (see example 4227).

• Using multiplicative change to find total effect of sequential changes: To determine the total effect after multiple changes, multiply all individual change factors. This approach simplifies calculations involving successive percentage increases or decreases, as demonstrated in examples 4227, 4234, and 4236.

📝 Essential Points

• The total change after a series of modifications is obtained by multiplying the individual change factors, not by adding percentage changes directly (see example 4227).

• When changes involve percentages, convert them into change factors: for an increase of p%, the change factor is 1 + p/100; for a decrease, it is 1 - p/100.

• The order of applying changes matters when dealing with percentages or change factors, especially in cases involving discounts or price reductions (see example 4234).

• To find the final value after multiple sequential changes, multiply the initial value by the total change factor, which is the product of all individual change factors.

💡 Key Takeaway

Multiplicative change allows us to efficiently calculate the combined effect of sequential percentage increases or decreases by multiplying their respective change factors, providing a straightforward method to analyze complex series of changes.

📖 7. Population Growth

🔑 Key Concepts & Definitions

• Population growth modeling: The process of representing how a population changes over time, often using percentage increases or decreases to predict future population sizes (see example calculations in exercises 4226, 4228, 4236).

• Calculating population after percentage increase over years: To find the population after a certain number of years with a consistent percentage increase, multiply the initial population by the growth factor raised to the power of the number of years. The growth factor for a percentage increase $p\%$ is $1 + \frac{p}{100}$ (see exercises 4226, 4228, 4236).

• Total percentage increase in population over multiple years: The overall change in population after several years with different percentage increases is found by multiplying the individual growth factors for each year or period, then subtracting 1 to find the total percentage increase (see exercises 4226, 4236).

📝 Essential Points

• The growth factor corresponding to a percentage increase $p\%$ is $1 + \frac{p}{100}$. For example, a 10% increase has a growth factor of 1.10, and a 20% increase has a growth factor of 1.20.

• When calculating the population after multiple periods with percentage increases, multiply the initial population by each growth factor sequentially. For example, if a population increases by 10% first and then 20%, the total growth factor is $1.10 \times 1.20 = 1.32$, meaning a total increase of 32%.

• To find the total percentage increase over multiple years, subtract 1 from the total growth factor and multiply by 100. For example, $(1.10 \times 1.20 - 1) \times 100 = 32\%$.

• The exercises demonstrate practical applications, such as predicting restaurant bookings (4226), tracking personal records (4228), and calculating population growth over years (4236).

💡 Key Takeaway

Understanding how to model population growth using percentage increases and growth factors allows for accurate predictions of future populations and analysis of cumulative changes over multiple periods.

📖 8. Tax and Discount Application

🔑 Key Concepts & Definitions

• Application of tax (VAT) after discount: The process of calculating the final price by first applying discounts to the original price, then adding VAT (Value Added Tax) to the reduced amount. This method ensures VAT is only charged on the net price after discounts (see concepts related to sequential discount and tax application).

• Calculating final price with sequential discount and tax: A step-by-step process where multiple discounts are applied successively to the original price, followed by the addition of tax to the discounted price. The order of applying discounts and tax significantly influences the final amount (see the effect of order of applying discount and tax).

• Effect of order of applying discount and tax on final price: The sequence in which discounts and taxes are applied affects the total payable amount. Applying discounts before tax generally results in a lower final price compared to applying tax first, due to the tax being calculated on a smaller base (see the example calculations and theoretical explanations).

📝 Essential Points

• When discounts are applied sequentially, the total discount factor is the product of individual discount factors, i.e., $(1 - d_1) \times (1 - d_2) \times \dots$, where $d_i$ are discount rates in decimal form.

• The application of VAT after discounts involves calculating the discounted price first, then multiplying by $(1 + \text{VAT rate})$ to include tax. This is the standard method in many countries and ensures VAT is only charged on the net amount.

• The order of applying discounts and VAT impacts the final price:

  • Discount then VAT: Final price = (Original price × discount factors) × (1 + VAT rate).
  • VAT then discount: Final price = Original price × (1 + VAT rate) × discount factors (which generally results in a higher final price).

• For multiple discounts, the combined discount factor is less than the sum of individual discounts, emphasizing the importance of sequential calculations.

💡 Key Takeaway

The sequence of applying discounts and tax significantly influences the final price; applying discounts before VAT typically results in a lower total payable amount, highlighting the importance of understanding the order of operations in pricing calculations.

📊 Synthesis Tables

⚠️ Common Pitfalls & Confusions

1. Adding percentage increases or discounts instead of multiplying change factors.
2. Confusing the change factor (e.g., 1.10) with the percentage increase (10%).
3. Applying discounts additively rather than multiplicatively, leading to overestimation of savings.
4. Forgetting to convert percentages to decimal form before calculations.
5. Assuming linear growth in compound growth scenarios; growth is exponential.
6. Misinterpreting successive percentage decreases as simple subtraction from 100%.
7. Ignoring the order of applying multiple discounts or percentage increases, which affects the final result.

✅ Exam Checklist

• Know the definition and calculation of percentage increase factors, including how to convert a percentage increase into a multiplier.
• Understand how to apply multiple percentage increases sequentially by multiplying their change factors.
• Be able to calculate compound growth over multiple periods using growth factors.
• Know how to compute the final price after successive discounts by multiplying discount factors.
• Understand the concept of investment growth over time, including applying monthly or periodic percentage changes.
• Recognize that applying multiple discounts or percentage increases is multiplicative, not additive.
• Be familiar with the formula for compound growth and how to use it in finance and population studies.
• Know SMITH's definition of the invisible hand (if relevant to the course content).
• Understand how to calculate the total effect of successive percentage decreases or increases in investment or prices.
• Be able to identify and avoid common pitfalls such as adding percentages instead of multiplying change factors.
• Master the calculation of overall percentage change from multiple sequential changes.
• Recall key examples and formulas related to each topic for quick application during the exam.