# IGCSE Mathematics Foundation: Percentages

## Scheme of work: IGCSE Foundation: Year 10: Term 4: Percentages

#### Prerequisite Knowledge

1. Converting Between Fractions, Decimals, and Percentages
• Concept: Understanding how to convert between fractions, decimals, and percentages.
• Example: $\text{Convert } \frac{1}{4} \text{ to a decimal: } \frac{1}{4} = 0.25.$ $\text{Convert 0.25 to a percentage: } 0.25 \times 100 = 25\%.$ $\text{Convert 25% to a fraction: } 25% = \frac{25}{100} = \frac{1}{4}.$
2. Finding Equivalent Fractions
• Concept: Understanding how to find fractions that are equivalent to a given fraction by multiplying or dividing the numerator and the denominator by the same number.
• Example: $\text{Equivalent fractions for } \frac{1}{2} \text{ are } \frac{2}{4}, \frac{3}{6}, \text{ and } \frac{4}{8}.$ $\text{For } \frac{3}{4}, \text{ multiplying numerator and denominator by 2 gives } \frac{6}{8}.$

#### Success Criteria

1. Express a Given Number as a Percentage of Another Number
• Objective: Calculate the percentage that one number represents of another number.
• Example: $\text{Express 30 as a percentage of 50.}$ $\frac{30}{50} \times 100 = 60\%.$
2. Understand the Multiplicative Nature of Percentages as Operators
• Objective: Use percentages as multiplicative operators to find a percentage of a quantity.
• Example: $\text{Find 25\% of 80.}$ $0.25 \times 80 = 20.$
3. Solve Simple Percentage Problems, Including Percentage Increase and Decrease
• Objective: Calculate the result of a percentage increase or decrease.
• Example (Increase): $\text{Increase 50 by 20\%.}$ $50 + (0.20 \times 50) = 50 + 10 = 60.$
• Example (Decrease): $\text{Decrease 80 by 25\%.}$ $80 – (0.25 \times 80) = 80 – 20 = 60.$
4. Use Reverse Percentages
• Objective: Find the original amount given the final amount after a percentage increase or decrease.
• Example: $\text{If 120 is 150\% of the original amount, find the original amount.}$ $\frac{120}{1.5} = 80.$
5. Use Compound Interest and Depreciation
• Objective: Calculate the compound interest or depreciation over a period of time.
• Example (Compound Interest): $\text{Calculate the amount after 2 years if } \1000 \text{ is invested at an annual interest rate of 5\%.}$ $A = P(1 + r/n)^{nt} = 1000(1 + 0.05/1)^{1 \times 2} = 1000(1.05)^2 = 1102.50.$
• Example (Depreciation): $\text{Calculate the value of an asset after 3 years if it depreciates at an annual rate of 10\% and its initial value is } \2000.$ $A = P(1 – r/n)^{nt} = 2000(1 – 0.10/1)^{1 \times 3} = 2000(0.90)^3 = 1458.$

#### Key Concepts

1. Express a Given Number as a Percentage of Another Number
• Concept: Understanding that to find the percentage one number represents of another, you divide the first number by the second number and multiply by 100.
2. Understand the Multiplicative Nature of Percentages as Operators
• Concept: Recognizing that percentages can be used as multiplicative operators to find a part of a whole. This involves converting the percentage to a decimal and multiplying by the quantity.
3. Solve Simple Percentage Problems, Including Percentage Increase and Decrease
• Concept: Percentage increase or decrease involves finding the percentage of the original amount and then adding or subtracting it from the original amount.
4. Use Reverse Percentages
• Concept: Reverse percentages involve finding the original amount before a percentage increase or decrease. This is done by dividing the final amount by the percentage factor.
5. Use Compound Interest and Depreciation
• Concept: Compound interest and depreciation involve repeated application of a percentage increase or decrease over multiple periods. This can be calculated using the formula for compound interest and depreciation.

#### Common Misconceptions

1. Express a Given Number as a Percentage of Another Number
• Common Mistake: Incorrectly setting up the fraction or forgetting to multiply by 100.
• Example: $\text{Incorrect: } \frac{30}{50} = 0.6 \text{ without converting to a percentage.}$ $\text{Correct: } \frac{30}{50} \times 100 = 60\%.$
2. Understand the Multiplicative Nature of Percentages as Operators
• Common Mistake: Using the percentage as a whole number instead of converting it to a decimal before multiplication.
• Example: $\text{Incorrect: Calculating 25\% of 80 as } 25 \times 80.$ $\text{Correct: Calculating 25\% of 80 as } 0.25 \times 80 = 20.$
3. Solve Simple Percentage Problems, Including Percentage Increase and Decrease
• Common Mistake: Misinterpreting the percentage increase or decrease, leading to incorrect calculations.
• Example (Increase): $\text{Incorrect: Increasing 50 by 20\% as } 50 + 20 = 70.$ $\text{Correct: Increasing 50 by 20\% as } 50 + (0.20 \times 50) = 60.$
• Example (Decrease): $\text{Incorrect: Decreasing 80 by 25\% as } 80 – 25 = 55.$ $\text{Correct: Decreasing 80 by 25\% as } 80 – (0.25 \times 80) = 60.$
4. Use Reverse Percentages
• Common Mistake: Incorrectly setting up the reverse percentage calculation, often by not using the correct percentage factor.
• Example: $\text{Incorrect: If 120 is 150\% of the original amount, finding the original amount by dividing 120 by 150.}$ $\text{Correct: If 120 is 150\% of the original amount, the original amount is } \frac{120}{1.5} = 80.$
5. Use Compound Interest and Depreciation
• Common Mistake: Misapplying the compound interest or depreciation formula, particularly by using incorrect values for the rate or number of periods.
• Example (Compound Interest): $\text{Incorrect: Calculating the amount after 2 years with a 5\% annual interest rate on } \1000 \text{ as } 1000 \times 0.05 \times 2.$ $\text{Correct: Using the formula } A = P(1 + r/n)^{nt} = 1000(1.05)^2 = 1102.50.$
• Example (Depreciation): $\text{Incorrect: Calculating the value of an asset after 3 years with a 10\% annual depreciation rate on } \2000 \text{ as } 2000 \times 0.10 \times 3.$ $\text{Correct: Using the formula } A = P(1 – r/n)^{nt} = 2000(0.90)^3 = 1458.$

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