Mathematics (0580) | Money

💰 IGCSE Mathematics Study Notes: The Money Chapter 💰

Hello future financial wizards! Welcome to the essential guide for the 'Money' topic in IGCSE Maths. This chapter is one of the most practical parts of your syllabus, focusing on how we handle money, rates, and percentages in the real world—from banking to budgeting.

Don't worry if words like 'compound interest' sound intimidating; we will break them down into simple, manageable steps. By the end of these notes, you'll be confident calculating discounts, converting currencies, and understanding your savings!

1. Calculations and Currency Conversion (C1.15 / E1.15)

1.1 Calculating with Money

Money calculations usually involve the four basic operations (+, –, ×, ÷), but they have a few special rules you must follow for accuracy, especially when using a calculator (C1.13 / E1.13).

Key Rules for Money

• Standard Format: Money is usually written with two decimal places. If your calculator shows 4.8, you must write it as $4.80.
• Rounding: Always round your final answer only to the nearest cent/penny (two decimal places), unless the question asks otherwise. Do not round intermediate steps!
• Interpreting Results: If a calculation results in an amount like 3.25 hours, in the context of money, make sure you interpret it correctly (C1.14 / E1.14), but remember 4.8 means $4.80.

Common Mistake to Avoid: Rounding during the middle of a multi-step calculation. This leads to an inaccurate final answer. Use the full, unrounded value in your calculator for subsequent steps.

Key Takeaway: Always present your monetary answers using two decimal places and ensure all calculations use the highest possible accuracy.

1.2 Currency Exchange Rates (C1.11 / E1.11 & C1.15 / E1.15)

An Exchange Rate is simply a rate that tells you how much one unit of currency is worth in another currency. This is a common practical application of rates.

Step-by-Step Conversion

The biggest challenge is knowing whether to multiply or divide. Think about the value you are starting with versus the value you are going to.

Imagine the exchange rate is: 1 US Dollar (USD) = 0.85 Euros (EUR).

Case 1: Converting the larger unit to the smaller unit (USD to EUR)
If you have $500, you expect to get a smaller number of Euros (because 1 USD is worth less than 1 EUR in this example).
Method: Multiply by the rate.
$$ 500 \text{ USD} \times 0.85 = 425 \text{ EUR} $$

Case 2: Converting the smaller unit to the larger unit (EUR to USD)
If you have 425 EUR, you expect to get a larger number of dollars (because 1 EUR buys more USD than 1 USD buys EUR).
Method: Divide by the rate.
$$ 425 \text{ EUR} \div 0.85 = 500 \text{ USD} $$

💡 Memory Aid: If the rate gives you '1 unit of A' = 'X units of B', use multiplication to go from A to B, and division to go from B to A.

Key Takeaway: Currency conversion is a problem of proportion. Always check if your answer makes sense—if you convert strong currency to weak currency, the numerical value should increase.

2. Percentage Calculations in Finance (C1.12 / E1.12)

Percentages are used everywhere in finance, from sale prices to calculating profit. You need to be able to calculate percentages, express quantities as percentages, and calculate changes.

2.1 Percentage of a Quantity (Deposit, Discount, Earnings)

To find a percentage of an amount, convert the percentage into a decimal or a fraction (usually a decimal is faster).

Example: Find a 15% discount on a $80 shirt.
Step 1: Convert 15% to a decimal: 0.15.
Step 2: Multiply the amount by the decimal.
$$ 0.15 \times 80 = 12 $$
The discount is $12.
Step 3: Calculate the final cost (if required).
$$ 80 - 12 = 68 $$
The final cost is $68.00.

2.2 Expressing One Quantity as a Percentage of Another

This is useful for finding the percentage profit or loss you made compared to the original cost.

Formula: $$ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% $$

Example: A shop buys a jacket for $50 and sells it for $75. What is the percentage profit?
Step 1: Calculate the profit (the 'Part').
$$ \text{Profit} = 75 - 50 = 25 $$
Step 2: Use the original cost (the 'Whole') to calculate the percentage.
$$ \text{Percentage Profit} = \frac{25}{50} \times 100\% = 50\% $$

2.3 Percentage Increase and Decrease (Percentage Change)

This method uses multipliers and is highly efficient.

• Increase: Add the percentage to 100%. (e.g., 20% increase = 120% = multiplier of 1.20)
• Decrease (Discount/Loss): Subtract the percentage from 100%. (e.g., 30% discount = 70% = multiplier of 0.70)

Example: A house worth $200 000 increases in value by 8%.
Multiplier: \( 100\% + 8\% = 108\% = 1.08 \)
$$ 200\ 000 \times 1.08 = 216\ 000 $$
The new value is $216 000.

Key Takeaway: Multipliers simplify percentage change calculations dramatically. An increase uses a multiplier > 1; a decrease uses a multiplier < 1.

3. Interest Calculations (C1.12 / E1.12)

Interest is the money earned on an investment (deposit) or the cost paid on a loan. The syllabus requires calculations involving Simple Interest and Compound Interest.

3.1 Simple Interest

Simple interest means the interest is calculated only on the original amount (the principal). The interest amount is the same every year.

Method: Calculate the interest for one period, then multiply by the total number of periods.

Example: $1000 is invested for 3 years at 5% simple interest per year.
Step 1: Calculate interest for 1 year (5% of $1000).
$$ 0.05 \times 1000 = 50 $$
Step 2: Calculate total interest for 3 years.
$$ 50 \times 3 = 150 $$
Total Value: \( 1000 + 150 = 1150 \)

3.2 Compound Interest

Compound interest is sometimes called 'interest on interest'. After the first period, the interest earned is added to the principal, and the next period's interest is calculated on this larger total.

Analogy: Simple interest is like earning interest on a single seed. Compound interest is like earning interest on a snowball—it gets bigger every year, so the interest earned also grows!

We use the multiplier method (repeated percentage change) for compound interest.

General Form (Method based on formula, which is not provided in exams): $$ \text{Final Amount} = \text{Principal} \times (\text{Multiplier})^{\text{Number of Years}} $$

Example: $5000 is invested for 4 years at 3% compound interest per year.
Step 1: Determine the multiplier (3% increase).
$$ 100\% + 3\% = 103\% = 1.03 $$
Step 2: Apply the multiplier for the number of years (4).
$$ \text{Final Amount} = 5000 \times (1.03)^4 $$
$$ 5000 \times 1.12550881 \approx 5627.544... $$
Step 3: Round the final amount to two decimal places.
Final amount = $5627.54.

Tip: If the question asks for the Total Interest Earned, remember to subtract the original principal from the final amount.
\( \text{Total Interest} = \$5627.54 - \$5000 = \$627.54 \)

Key Takeaway: Compound interest uses the power of multipliers. If you use the multiplier method correctly, you directly calculate the final amount.

4. Extended Content: Reverse Percentages (E1.12)

This section is crucial for students taking the Extended syllabus. Reverse percentages (or 'working backwards') are required when you are given the final amount *after* a percentage change and you need to find the original amount.

4.1 The Reverse Percentage Method

We still use the multiplier, but we use division instead of multiplication because we are working in reverse.

Step-by-Step Reverse Percentage

Example: A dress is sold for $144 after a 20% discount. What was the original price?

Common Mistake: Calculating 20% of $144 and adding it back. This is wrong because the 20% discount was based on the original price, not the sale price.

Step 1: Find the percentage the final price represents.
A 20% discount means the sale price is \( 100\% - 20\% = 80\% \) of the original price.

Step 2: Determine the multiplier.
Multiplier = 0.80.

Step 3: Set up the equation and solve using division.
$$ \text{Original Price} \times 0.80 = 144 $$ $$ \text{Original Price} = \frac{144}{0.80} = 180 $$
The original price was $180.00.

Example 2 (Profit/VAT): A painter makes a 15% profit on a painting, selling it for $230. How much did the painter pay for it (cost price)?
Step 1: The selling price represents \( 100\% + 15\% = 115\% \) of the cost price.
Step 2: Multiplier = 1.15.
Step 3: Divide the final amount by the multiplier.
$$ \text{Cost Price} = \frac{230}{1.15} = 200 $$
The cost price was $200.00.

Key Takeaway: When working backwards (reverse percentages), always divide the known quantity by the appropriate multiplier to find the original 100% value.

5. Other Practical Rates (C1.11 / E1.11)

The 'Money' concepts often overlap with the general 'Rates' section of the syllabus, which deals with quantities expressed per unit of another quantity.

5.1 Hourly Rates of Pay (Earnings)

This involves calculating total earnings based on an hourly wage and the number of hours worked.

Example: If you earn $12.50 per hour and work 35 hours, your total earnings are:
$$ 12.50 \times 35 = 437.50 $$
Total earnings = $437.50.

5.2 Fuel Consumption

This measures the efficiency of a vehicle, often given in litres per 100 km (L/100 km).

Example: A car uses 8 litres of fuel to travel 100 km. If fuel costs $1.50 per litre, calculate the fuel cost for a 500 km journey.

Step 1: Find the total fuel needed for 500 km. Since 500 km is five times 100 km:
$$ 8 \text{ L/100 km} \times 5 = 40 \text{ Litres} $$
Step 2: Calculate the total cost.
$$ 40 \times 1.50 = 60 $$
Total fuel cost = $60.00.

Did you know? Understanding rates and percentages helps determine the "best value" when shopping. You can use the formula: $\text{Price} \div \text{Quantity}$ to find the unit price, helping you decide which size product is the cheapest per kilogram or per item.

Key Takeaway: Rates are simply proportions. Set up the conversion logically to ensure you multiply or divide correctly to find the required quantity, whether it's kilometers, hours, or money.