CIE IGCSE Mathematics (0580) Percentages Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Percentages Study Notes
LEARNING OBJECTIVE
- Concepts of Percentage
Key Concepts:
- Percentage
- Simple Interest
- Compound Interest
Percentage
Calculating a Given Percentage of a Quantity
Percentages are used to describe parts of a whole. To calculate a percentage of a given quantity, convert the percentage into a fraction or decimal and multiply it by the quantity.
This process is useful in many real-life situations, such as finding discounts, tax amounts, or commissions.
\( \text{Percentage of a Quantity} = \frac{\text{Percentage}}{100} \times \text{Quantity} \)
Example:
Find 35% of $240.
▶️ Answer/Explanation
\( \frac{35}{100} \times 240 = 0.35 \times 240 = 84 \)
Answer: $84
Expressing One Quantity as a Percentage of Another
This method allows you to compare two quantities in percentage terms, showing how much one value is relative to another.
This is especially helpful when analyzing marks, population growth, prices, and proportions.
\( \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \)
Example:
Express 36 as a percentage of 60.
▶️ Answer/Explanation
\( \frac{36}{60} \times 100 = 0.6 \times 100 = 60\% \)
Answer: 60%
Percentage Increase or Decrease
Percentage change measures how much a quantity has increased or decreased compared to its original value.
An increase occurs when the new value is greater than the original, and a decrease occurs when it is less.
\( \text{Percentage Change} = \left( \frac{\text{New} – \text{Original}}{\text{Original}} \right) \times 100 \)
This is commonly used in price changes, sales analysis, population comparisons, etc.
Example:
A jacket was priced at $80 but now costs $100. What is the percentage increase?
▶️ Answer/Explanation
\( \frac{100 – 80}{80} \times 100 = \frac{20}{80} \times 100 = 25\% \)
Answer: 25% increase
Simple & Compound Interest
Simple Interest
Simple interest is calculated only on the original principal amount for a given time at a fixed rate of interest.
It is commonly used in short-term loans and straightforward financial agreements.
Formula: \( \text{Simple Interest} = \frac{P \times R \times T}{100} \)
Where:
\( P \) = Principal (initial amount)
\( R \) = Rate of interest per annum (%)
\( T \) = Time in years
Compound Interest
Compound interest is calculated on the initial principal and also on the interest that has been added over previous periods. It is often used in savings accounts, loans, and investments where interest is “compounded” annually or more frequently.
Formula for total amount: \( \text{Amount} = P \left(1 + \frac{R}{100}\right)^T \)
Compound Interest = \( \text{Amount} – P \)
Other Percentage Calculations:
- Deposit: A percentage of a total cost paid upfront. The remainder is usually paid later or borrowed (e.g. $20\%$ deposit on a $\$5000$ item is $\$1000$).
- Discount: A reduction in price by a given percentage, often during sales.
- Profit and Loss: Determined by comparing the selling price (SP) and cost price (CP). Profit occurs if SP > CP, and loss if SP < CP. Both can be expressed as absolute amounts or percentages.
- Earnings: Includes commissions, bonuses, and other pay calculated as a percentage of sales or base pay.
- Percentages Over 100%: Used when values grow beyond their original amounts (e.g. 160% means 60% more than the original).
Example:
Calculate the simple interest on $800 at 6% per annum for 3 years.
▶️ Answer/Explanation
\( \text{SI} = \frac{800 \times 6 \times 3}{100} = \frac{14400}{100} = 144 \)
Answer: $144
Example:
Find the compound interest on $1000 at 5% per annum for 2 years.
▶️ Answer/Explanation
\( \text{Amount} = 1000 \left(1 + \frac{5}{100}\right)^2 = 1000 \times (1.05)^2 = 1000 \times 1.1025 = 1102.50 \)
\( \text{CI} = 1102.50 – 1000 = 102.50 \)
Answer: $102.50
Example:
A shopkeeper buys a phone for $\$300$ and sells it for $\$390$. What is the percentage profit?
▶️ Answer/Explanation
Profit = \( 390 – 300 = 90 \)
Profit% = \( \frac{90}{300} \times 100 = 30\% \)
Answer: 30% profit
Example:
Priya buys a second-hand car costing $\$6000$. She pays a 25% deposit and borrows the rest from a bank at 8% simple interest per annum for 2 years. After 2 years, she sells the car for $\$6200$.
Calculate:
(a) The amount borrowed from the bank
(b) The interest paid
(c) Her total repayment
(d) Her profit or loss
▶️ Answer/Explanation
(a) Amount borrowed:
Deposit = \( \frac{25}{100} \times 6000 = 1500 \)
Borrowed = \( 6000 – 1500 = 4500 \)
(b) Interest paid:
\( \text{SI} = \frac{4500 \times 8 \times 2}{100} = \frac{72000}{100} = 720 \)
(c) Total repayment to bank:
\( 4500 + 720 = 5220 \)
(d) Profit or loss:
Total cost = \( 1500 + 5220 = 6720 \)
Selling price = $6200
Loss = \( 6720 – 6200 = 520 \)
Answer: She made a loss of $520