Percentages

1. Overview

Percentages are a fundamental tool in mathematics for expressing proportions and comparing quantities relative to a whole (100). They are crucial for various real-world applications, including financial calculations (interest, discounts, taxes), statistics, and interpreting data. This revision guide covers calculating percentages, percentage changes, and working with interest, equipping you with the skills needed for IGCSE Mathematics exams.

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Key Definitions

• Percentage: A fraction with a denominator of 100. The symbol % means "per hundred."
• Multiplier: A decimal value used to calculate a percentage change in one step (e.g., an increase of 5% uses a multiplier of 1.05).
• Principal ($P$): The original amount of money invested or borrowed.
• Simple Interest: Interest calculated only on the original principal amount throughout the duration of the loan or investment.
• Compound Interest: Interest calculated on the principal plus any interest accumulated from previous periods ("interest on interest").

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Core Content

3.1 Calculate a Given Percentage of a Quantity

To find a percentage of an amount, convert the percentage to a decimal or fraction and multiply.

Worked example 1 — Finding a percentage of a value

Question: Calculate 32% of US$150.

1. Convert the percentage to a decimal: $32% = \frac{32}{100} = 0.32$
   • This converts the percentage into a usable decimal form.
2. Multiply the decimal by the quantity: $0.32 \times 150 = 48$
   • This calculates the desired percentage of the total amount.

• Final Answer: $\boxed{$48}$

Worked example 2 — Finding a percentage of a length

Question: What is 65% of 240 cm?

1. Convert the percentage to a decimal: $65% = \frac{65}{100} = 0.65$
   • Divide the percentage by 100 to get its decimal equivalent.
2. Multiply the decimal by the length: $0.65 \times 240 = 156$
   • This calculates the desired percentage of the total length.

• Final Answer: $\boxed{156 \text{ cm}}$

3.2 Express One Quantity as a Percentage of Another

Write the "part" over the "whole" as a fraction, then multiply by 100.

Worked example 3 — Expressing one quantity as a percentage of another

Question: Express 27 as a percentage of 75.

1. Write as a fraction: $\frac{27}{75}$
   • This represents the ratio of the part to the whole.
2. Multiply by 100: $\frac{27}{75} \times 100$
   • This converts the fraction to a percentage.
3. Simplify: $0.36 \times 100 = 36$
   • Calculate the result.

• Final Answer: $\boxed{36%}$

Worked example 4 — Percentage of students

Question: In a class of 30 students, 12 are girls. What percentage of the class are girls?

1. Write as a fraction: $\frac{12}{30}$
   • Express the number of girls as a fraction of the total number of students.
2. Multiply by 100: $\frac{12}{30} \times 100$
   • Convert the fraction to a percentage.
3. Simplify: $0.4 \times 100 = 40$
   • Calculate the result.

• Final Answer: $\boxed{40%}$

3.3 Calculate Percentage Increase or Decrease

To find the percentage change: $\frac{\text{Difference (New - Old)}}{\text{Original Amount}} \times 100$.

Worked example 5 — Calculating percentage decrease

Question: A bicycle's price decreased from $160 to $120. Calculate the percentage decrease.

1. Find the difference: $160 - 120 = 40$
   • This calculates the amount of the decrease.
2. Divide by the original price: $\frac{40}{160}$
   • This expresses the decrease as a fraction of the original price.
3. Multiply by 100: $\frac{40}{160} \times 100 = 25$
   • Convert the fraction to a percentage.

• Final Answer: $\boxed{25%}$

Worked example 6 — Calculating percentage increase

Question: The population of a town increased from 15,000 to 18,000. Calculate the percentage increase.

1. Find the difference: $18,000 - 15,000 = 3,000$
   • This calculates the amount of the increase.
2. Divide by the original population: $\frac{3,000}{15,000}$
   • This expresses the increase as a fraction of the original population.
3. Multiply by 100: $\frac{3,000}{15,000} \times 100 = 20$
   • Convert the fraction to a percentage.

• Final Answer: $\boxed{20%}$

3.4 Simple and Compound Interest

Simple Interest ($I$): Calculated using $I = \frac{PRT}{100}$.

Worked Example 7 (Simple Interest): Question: Calculate the total amount after $800 is invested for 5 years at 3% simple interest per year.

1. Identify variables: $P = 800, R = 3, T = 5$.
   • State the given values.
2. Calculate interest ($I$): $I = \frac{800 \times 3 \times 5}{100} = 120$.
   • Substitute the values into the simple interest formula.
3. Calculate total amount: $800 + 120 = 920$.
   • Add the interest to the principal to find the total amount.

• Final Answer: $\boxed{$920}$
• Note: Always check if the question asks for the "interest" ($120) or the "total amount" ($920).

Compound Interest ($A$): Calculated using $A = P(1 + \frac{r}{100})^n$.

Worked Example 8 (Compound Interest): Question: Calculate the final value of $5000 invested for 4 years at 2.5% compound interest per year.

1. Identify variables: $P = 5000, r = 2.5, n = 4$.
   • State the given values.
2. Substitute into formula: $A = 5000 \times (1 + \frac{2.5}{100})^4$
   • Plug the values into the compound interest formula.
3. Simplify the multiplier: $A = 5000 \times (1.025)^4$
   • Simplify the expression inside the parentheses.
4. Calculate: $A = 5000 \times 1.103813 = 5519.065$
   • Calculate the final amount.

• Final Answer: $\boxed{$5519.07}$
• Note: In money questions, always include two decimal places if the value is not a whole number.

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Extended Content (Extended Only)

4.1 Reverse Percentages

This is used to find the original value after a percentage change has already occurred. Method: $\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}$

When dealing with reverse percentages, it's crucial to understand whether the given value represents an increase or a decrease from the original. If there's an increase, the multiplier will be greater than 1. If there's a decrease (discount), the multiplier will be less than 1. The key is to divide the new value by the correct multiplier to "undo" the percentage change and find the original value.

Worked Example 9 — Finding the original price after a discount

Question: A shop sells a television for $360 after applying a 20% discount. What was the original price of the television?

1. Determine the multiplier: A 20% discount means the television is sold at 80% of its original price ($100% - 20% = 80%$). Therefore, the multiplier is 0.80.
   • Calculate the percentage of the original price represented by the sale price.
2. Set up the equation: $\text{Original Price} \times 0.80 = 360$
   • Express the relationship between the original price, the multiplier, and the sale price.
3. Divide to find the original: $\text{Original Price} = \frac{360}{0.80} = 450$
   • Isolate the original price by dividing both sides of the equation by the multiplier.

• Final Answer: $\boxed{$450}$
• Warning: Do not simply calculate 20% of $360 and add it. This is a common error in IGCSE exams.

Worked Example 10 — Finding the original value after an increase

Question: After a 5% salary increase, John now earns $2520 per month. What was John's salary before the increase?

1. Determine the multiplier: A 5% increase means John's new salary is 105% of his original salary ($100% + 5% = 105%$). Therefore, the multiplier is 1.05.
   • Calculate the percentage of the original salary represented by the new salary.
2. Set up the equation: $\text{Original Salary} \times 1.05 = 2520$
   • Express the relationship between the original salary, the multiplier, and the new salary.
3. Divide to find the original: $\text{Original Salary} = \frac{2520}{1.05} = 2400$
   • Isolate the original salary by dividing both sides of the equation by the multiplier.

• Final Answer: $\boxed{$2400}$

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Key Equations

Simple Interest: $I = \frac{PRT}{100}$

• $I$: Interest earned ($)
• $P$: Principal (Original amount) ($)
• $R$: Rate of interest per year (%)
• $T$: Time (years)

Note: This formula is not provided on the IGCSE formula sheet; it must be memorised.

Compound Interest: $A = P(1 + \frac{r}{100})^n$

• $A$: Total amount after $n$ years ($)
• $P$: Principal (Original amount) ($)
• $r$: Rate of interest per year (%)
• $n$: Number of years

Note: This formula is not provided on the IGCSE formula sheet; it must be memorised.

Percentage Change: $\text{Percentage Change} = \frac{\text{Change}}{\text{Original}} \times 100$

Note: This formula is not provided on the IGCSE formula sheet; it must be memorised.

Reverse Percentage: $\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}$

Note: This formula is not provided on the IGCSE formula sheet; it must be memorised.

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Common Mistakes to Avoid

• ❌ Wrong: Using the simple interest formula for a compound interest question. ✓ Right: Underline the words "simple" or "compound" in the question to ensure you use the correct method. Write down the correct formula ($I = \frac{PRT}{100}$ or $A = P(1 + \frac{r}{100})^n$) before you start.
• ❌ Wrong: Calculating 8% of a sale price to find the original price before an 8% discount. ✓ Right: Recognize this as a reverse percentage problem. If the price decreased by 8%, divide the sale price by 0.92 (1 - 0.08) to find the original price.
• ❌ Wrong: Rounding money to the nearest dollar (e.g., writing $25 instead of $24.58). ✓ Right: Keep money to exactly 2 decimal places unless it is a whole number. If your calculator gives you 24.576, round it to $24.58.
• ❌ Wrong: Giving a percentage answer as a single decimal (e.g., 7.2). ✓ Right: Unless specified, round percentages to 3 significant figures (e.g., 7.23%). Remember to include the % sign in your final answer.
• ❌ Wrong: Forgetting to subtract the principal when a question asks for the interest earned in a compound interest problem. ✓ Right: Read the question carefully. If it asks for the interest earned, calculate the final amount ($A$) using the compound interest formula, and then subtract the principal ($P$) from $A$ to find the interest.

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Exam Tips

• Command Words:
  • "Calculate...": Show every step of your multiplication/division.
  • "Express...": Ensure your final answer has a % sign.
• Read the prompt carefully: Does the question ask for the interest only or the total amount? This is a frequent "trap" that costs marks.
• Calculator Tip: For compound interest, use the power button ($x^y$ or $x^\square$) on your calculator for the years ($n$). Do not round the multiplier before raising it to the power, as premature rounding causes inaccuracies. Store the multiplier in your calculator's memory for even greater accuracy.
• Real-world contexts: Expect questions about bank accounts, population growth (usually compound), and shop sales (discounts).
• Check logic: In a discount question, your answer (the original price) must be higher than the sale price. In an increase question, your answer (the original value) must be lower than the final value.
• Units: Always include appropriate units in your final answer (e.g., $, cm, kg).