International Mathematics (0607) | Money

💰 IGCSE International Mathematics 0607: Study Notes on Money 💰

Hello future Mathematician! Welcome to the "Money" chapter—one of the most practical and important topics in your IGCSE course. Why? Because the skills you learn here—calculating interest, dealing with exchange rates, and understanding profit—are used every single day by real people and businesses around the world!

This chapter is part of the Number section. We will be focusing on calculating with money and applying percentages and rates to financial situations, ensuring you can manage money problems like a pro!

1. Essential Calculations with Money (C1.15.1 / E1.15.1)

Calculating with money is fundamentally about using the four operations (addition, subtraction, multiplication, division) correctly, especially when dealing with decimals and context.

Key Terms in Business Math

• Cost Price (CP): The price a shop or person pays for an item.
• Selling Price (SP): The price the item is sold for.
• Profit: When SP > CP.
• Loss: When CP > SP.
• Discount: A reduction in price, usually expressed as a percentage.
• Earnings: The money received from work or investments.

1.1 Profit, Loss, and Discount

The core formulas are simple:

$$ \text{Profit} = \text{Selling Price} - \text{Cost Price} $$
$$ \text{Loss} = \text{Cost Price} - \text{Selling Price} $$

We often need to calculate the percentage profit or loss, which is always based on the original cost price:

$$ \text{Percentage Profit/Loss} = \frac{\text{Change in Price}}{\text{Original Cost Price}} \times 100 $$

Example: A shop buys a phone for $200 (CP) and sells it for $250 (SP).
Profit = $250 - $200 = $50.
Percentage Profit = \( \frac{50}{200} \times 100 = 25\%\).

2. Currency Conversion (C1.15.2 / E1.15.2)

When you travel or buy things internationally, you need to change one currency into another. This uses an Exchange Rate, which is a type of ratio or rate (C1.11/E1.11).

2.1 Understanding Exchange Rates

An exchange rate tells you how much of one currency you get for one unit of another.
Example: 1 USD = 0.85 EUR

The Golden Rule: Multiply or Divide?

This is the part students often confuse! Here is a simple trick:

• Going to the smaller number? DIVIDE.

  Example: Converting a lot of Euros into 1 USD. You divide the EUR amount by 0.85 to find the number of USD you get.

• Going to the bigger number? MULTIPLY.

  Example: Converting 1 USD into a lot of Euros. You multiply the USD amount by 0.85.

Don't worry if this seems tricky at first. Always set up the problem with the exchange rate first, then think logically:

Example: Exchange rate is 1 GBP = 1.30 USD. You have 500 GBP. How many USD do you get?

1. Start with what you have: 500 GBP.
2. We are converting 1 GBP into a larger amount (1.30 USD).
3. Action: Multiply.
4. Calculation: \( 500 \times 1.30 = 650 \) USD.

Example 2: The rate is 1 USD = 110 JPY (Japanese Yen). You want to convert 55,000 JPY back into USD.

1. Start with what you have: 55,000 JPY.
2. We are converting many Yen into 1 USD.
3. Action: Divide.
4. Calculation: \( 55000 \div 110 = 500 \) USD.

3. Financial Growth: Interest (C1.12.4 / E1.12.4)

When you borrow money (like a loan) or save money (like a deposit), interest is involved. It is essentially the cost of borrowing or the reward for saving.

⚠️ IMPORTANT: The syllabus states that formulas for Simple and Compound Interest are NOT GIVEN in Papers 1–4. You must memorise these structures!

3.1 Simple Interest (The Easy Way)

Simple interest is calculated only on the original amount (the principal). The amount of interest earned each period remains constant.

• P = Principal (the starting amount)
• R = Rate (annual interest rate as a percentage)
• T = Time (in years)

The formula for calculating the total Interest (I) earned is:

$$ I = \frac{P \times R \times T}{100} $$

The Total Amount (A) after T years is:

$$ A = P + I $$

Example: $1000 is invested at 5% simple interest for 3 years.

$$ I = \frac{1000 \times 5 \times 3}{100} = 150 $$

The total amount is $1000 + $150 = $1150.

3.2 Compound Interest (The Snowball Effect)

Compound interest is interest calculated on the principal plus any accumulated interest from previous periods. Your money earns money, and the interest starts earning interest! This is much more powerful for saving.

• P = Principal (the starting amount)
• R = Rate (annual interest rate as a percentage)
• n = Number of compounding periods (usually years)
• A = Total Amount (Principal + Interest)

The formula for the Total Amount (A) is:

$$ A = P \left(1 + \frac{R}{100}\right)^n $$

If the question asks for the total interest earned, remember to subtract the original principal: \( I = A - P \).

Example: $1000 is invested at 5% compound interest for 3 years.

$$ A = 1000 \left(1 + \frac{5}{100}\right)^3 = 1000 (1.05)^3 $$ $$ A = 1000 \times 1.157625 = 1157.63 \text{ (to 2 decimal places)} $$

The total amount is $1157.63. (Notice this is more than the $1150 from simple interest!)

🚨 Common Mistake Alert

Students often confuse compound interest with repeated percentage change (like depreciation). While the structure is similar, ensure you use the correct rate (for depreciation, the rate factor is \(1 - \frac{R}{100}\)).

4. Reverse Percentage Problems (Extended Content E1.12.5)

These are often found in money questions, especially relating to sales tax (VAT) or profit, where you are given the final price and need to find the original price.

This calculation involves working backward from a known percentage change.

Analogy: Imagine you have a cake (the original price). The baker (the percentage change) has already eaten a slice. You see the remaining cake (the final price) and must figure out the size of the whole original cake.

Step-by-Step Reverse Percentage Method

1. Determine the Final Percentage: If the price increased by 20%, the final price is 120% of the original. If it decreased by 10% (a discount), the final price is 90% of the original.
2. Set up the Relationship: Relate the known price to its corresponding percentage.
3. Find 1%: Divide the final price by the final percentage value.
4. Find 100% (The Original Price): Multiply the value of 1% by 100.

Example: A shirt costs $72 after a 20% discount. What was the original price?

1. Final Percentage: $72 represents the original price minus 20% discount. So, $72 = 80%.
2. Set up: 80% = $72
3. Find 1%: \( 1\% = \frac{72}{80} = 0.90 \)
4. Find 100%: \( 100\% = 0.90 \times 100 = 90 \)

The original price was $90.

Alternatively, using Decimals (Multipliers)

1. If the original price is P, and the discount is 20%, then: \( P \times 0.80 = 72 \)
2. To find P, rearrange: \( P = \frac{72}{0.80} = 90 \)

5. Financial Interpretations and Accuracy

In exams, especially with money, accuracy is key.

1. Rounding (C1.13.3): Money is always rounded to two decimal places (e.g., $4.80, not 4.8 or 4.803). Even if your calculation gives you $12.33333..., the final answer must be $12.33.

2. Working (C1.13.1): Do not round intermediate steps! If you use a rounded figure in a later calculation, you may lose accuracy marks. Keep the full number in your calculator memory until the very final step.

3. Earnings and Rates (C1.11): Problems involving hourly rates of pay are simple multiplication (Rate per hour $\times$ Hours worked).

Example: If you earn $15 per hour and work 37 hours: \( 15 \times 37 = 555 \). Total earnings: $555.00.

Keep practising these core skills, and you will master the "Money" chapter quickly!