Welcome to Money and Financial Mathematics!
Managing money is one of the most important life skills you will ever learn. Whether you are calculating the discount on a new pair of trainers, figuring out your pay from a weekend job, or working out the best deal at the supermarket, financial mathematics is everywhere. Don't worry if maths hasn't always been your favourite subject — this chapter uses straightforward, real-world examples that will make complete sense!
Did you know? The concept of charging interest on borrowed money goes back over \(4000\) years to ancient Mesopotamia, where people borrowed grain and seeds and paid back extra after the harvest!
1. Everyday Money Basics and Best Buys
Writing Money Correctly
In the UK, our currency is pounds (\(£\)) and pence (\(p\)). There are \(100\text{ p}\) in \(£1\).
When writing money in pounds, always write your final answer to \(2\) decimal places, unless the question tells you otherwise.
• \(£4.5\) must always be written as \(£4.50\).
• \(£8.2\) must always be written as \(£8.20\).
• \(35\text{ p}\) can be written as \(35\text{ p}\) or \(£0.35\), but never mix the two symbols like \(£0.35\text{p}\)!
Comparing Offers: Value for Money ("Best Buys")
Supermarkets love offering different sizes and multipacks. To find out which option is the better value, you need to compare them fairly. There are two easy ways to do this:
Method 1: Find the Unit Cost (Cost per unit)
Divide the total cost by the quantity (e.g., cost per \(100\text{ g}\), cost per item, or cost per \(1\text{ ml}\)).
Formula: \(\text{Unit Cost} = \frac{\text{Total Price}}{\text{Quantity}}\)
The lower the unit cost, the better the value!
Method 2: Find the Quantity per Penny/Pound
Divide the quantity by the price.
Formula: \(\text{Quantity per } £1 = \frac{\text{Quantity}}{\text{Total Price}}\)
The higher the quantity you get for your money, the better the value!
Step-by-Step Example:
A \(500\text{ g}\) box of cereal costs \(£2.40\). A \(750\text{ g}\) box costs \(£3.45\). Which box gives you better value for money?
Step 1: Calculate the cost per gram (or per \(100\text{ g}\)) for each box.
• For the \(500\text{ g}\) box: \(\frac{£2.40}{500} = £0.0048\text{ per gram}\) (or \(£0.48\text{ per } 100\text{ g}\)).
• For the \(750\text{ g}\) box: \(\frac{£3.45}{750} = £0.0046\text{ per gram}\) (or \(£0.46\text{ per } 100\text{ g}\)).
Step 2: Compare the two numbers.
Since \(£0.0046 < £0.0048\), the \(750\text{ g}\) box is cheaper per gram.
Conclusion: The \(750\text{ g}\) box is the better value.
Common Mistake to Avoid: Don't just pick the cheapest item overall! Always compare the price relative to the amount you are getting.
Key Takeaway: Always bring both products to the same standard unit (like \(1\text{ g}\), \(100\text{ g}\), or \(1\text{ item}\)) before making a comparison.
2. Earnings: Gross Pay, Net Pay, and Overtime
Understanding Your Payslip
When you start working, your payslip will show several key terms:
• Gross Pay: The total amount of money you earn before any deductions (taxes, pensions) are taken out.
• Deductions: Money automatically taken from your pay, such as Income Tax and National Insurance.
• Net Pay (Take-Home Pay): The actual money that goes into your bank account.
Simple Formula:
\(\text{Net Pay} = \text{Gross Pay} - \text{Total Deductions}\)
Calculating Basic Pay and Overtime
• Basic Pay: \(\text{Hours Worked} \times \text{Hourly Rate}\)
• Overtime: Extra hours worked beyond standard hours, often paid at a higher rate like "Time and a half" (\(\times 1.5\)) or "Double time" (\(\times 2\)).
Step-by-Step Example:
Sam earns \(£10.00\) per hour for a standard \(35\text{-hour}\) week. Any overtime is paid at time and a half. In one week, Sam works \(40\) hours. Deductions total \(£48.50\). Calculate Sam's net pay.
Step 1: Calculate basic pay for standard hours.
\(\text{Basic Pay} = 35 \times £10.00 = £350.00\)
Step 2: Calculate overtime hours and overtime pay rate.
\(\text{Overtime Hours} = 40 - 35 = 5\text{ hours}\)
\(\text{Overtime Rate} = £10.00 \times 1.5 = £15.00\text{ per hour}\)
\(\text{Overtime Pay} = 5 \times £15.00 = £75.00\)
Step 3: Find total Gross Pay.
\(\text{Gross Pay} = £350.00 + £75.00 = £425.00\)
Step 4: Subtract deductions to find Net Pay.
\(\text{Net Pay} = £425.00 - £48.50 = £376.50\)
Key Takeaway: Gross is what you earn; Net is what you get. Remember: Net Pay is always smaller than Gross Pay.
3. Percentages in Finance: Profit, Loss, and Discounts
Profit and Loss
When businesses buy items to sell them, they want to make a profit.
• Cost Price (CP): The amount an item originally costs to buy or make.
• Selling Price (SP): The amount an item is sold for.
• \(\text{Profit} = \text{Selling Price} - \text{Cost Price}\) (when \(\text{SP} > \text{CP}\))
• \(\text{Loss} = \text{Cost Price} - \text{Selling Price}\) (when \(\text{CP} > \text{SP}\))
Percentage Profit or Loss Formula
\(\text{Percentage Profit or Loss} = \frac{\text{Actual Profit or Loss}}{\text{Original Cost Price}} \times 100\)
Memory Trick: Always put the change over the ORIGINAL cost price, never over the new selling price!
Step-by-Step Example:
A shopkeeper buys a bicycle for \(£160\) and sells it for \(£200\). What is the percentage profit?
Step 1: Calculate the actual profit.
\(\text{Profit} = £200 - £160 = £40\)
Step 2: Divide the profit by the original cost price and multiply by \(100\).
\(\text{Percentage Profit} = \frac{£40}{£160} \times 100 = 0.25 \times 100 = 25\%\)
Discounts and Multipliers
A discount is a reduction in price.
To find a discounted price quickly, use a decimal multiplier:
• A \(20\%\) discount means you pay \(100\% - 20\% = 80\%\). The multiplier is \(0.80\).
• A \(15\%\) discount means you pay \(100\% - 15\% = 85\%\). The multiplier is \(0.85\).
• Adding \(20\%\) VAT means you pay \(100\% + 20\% = 120\%\). The multiplier is \(1.20\).
Example: A jacket costs \(£80\). It is on sale with a \(15\%\) discount.
\(\text{Sale Price} = £80 \times 0.85 = £68.00\)
Key Takeaway: Using decimal multipliers saves time and avoids multiple calculation steps.
4. Interest: Simple vs. Compound
Interest is the extra money you earn when you save, or the extra fee you pay when you borrow.
A. Simple Interest
In simple interest, the interest earned is the exact same amount every year because it is calculated only on the original starting amount (the Principal).
The Simple Interest Formula:
\(I = \frac{P \times R \times T}{100}\)
Where:
• \(I\) = Total Interest earned/paid
• \(P\) = Principal (the starting amount of money)
• \(R\) = Rate of interest per year (as a percentage)
• \(T\) = Time (in years)
Example: You invest \(£1200\) for \(3\) years at an interest rate of \(4\%\) simple interest per year.
\(I = \frac{1200 \times 4 \times 3}{100} = \frac{14400}{100} = £144\)
\(\text{Total amount at the end} = \text{Principal} + \text{Interest} = £1200 + £144 = £1344.00\)
B. Compound Interest
Compound interest is where interest is added to your account, and then in the next year, you earn interest on your interest as well! This makes your money grow much faster.
The Compound Interest Formula:
\(\text{Total Amount} = P \times \left(1 + \frac{R}{100}\right)^n\)
Where:
• \(P\) = Principal (starting amount)
• \(R\) = Rate of interest per year (%)
• \(n\) = Number of years
Step-by-Step Example:
Maya invests \(£2500\) for \(3\) years at \(5\%\) compound interest per year. Calculate the total balance after \(3\) years.
Step 1: Find the multiplier.
\(100\% + 5\% = 105\% \implies 1.05\)
Step 2: Apply the formula using a power of \(3\) for the \(3\) years.
\(\text{Total Balance} = 2500 \times (1.05)^3\)
\(\text{Total Balance} = 2500 \times 1.157625 = £2894.0625\)
Step 3: Round correctly to \(2\) decimal places.
\(\text{Final Balance} = £2894.06\)
Depreciation (Compound Decrease)
Items like cars or mobile phones lose value over time. This is called depreciation. We use the same formula, but we subtract the percentage instead of adding it!
Formula: \(\text{Value} = P \times \left(1 - \frac{R}{100}\right)^n\)
Example: A car is bought for \(£14000\) and depreciates by \(12\%\) each year. What is its value after \(2\) years?
\(\text{Multiplier} = 100\% - 12\% = 88\% = 0.88\)
\(\text{Value} = £14000 \times (0.88)^2 = £14000 \times 0.7744 = £10841.60\)
Key Takeaway: Compound interest multiplies by the growth factor repeatedly using powers (\(n\)), while simple interest adds the exact same amount every year.
5. Foreign Exchange (Currency Conversions)
When travelling abroad, you exchange your money for foreign currency using an exchange rate.
The Core Rules:
• To convert Home Currency (\(£\)) \(\rightarrow\) Foreign Currency: MULTIPLY by the exchange rate.
• To convert Foreign Currency \(\rightarrow\) Home Currency (\(£\)): DIVIDE by the exchange rate.
Step-by-Step Examples:
Suppose the exchange rate is \(£1 = €1.16\) (Euros).
Example A: Convert \(£350\) into Euros (\(€\)).
Since we are going from pounds to foreign currency, we multiply:
\(£350 \times 1.16 = €406.00\)
Example B: A pair of sunglasses costs \(€58.00\) in Spain. How much is this in pounds (\(£\))?
Since we are going from foreign currency back to pounds, we divide:
\(€58.00 \div 1.16 = £50.00\)
Sanity Check Tip: Check if your answer makes sense. If \(£1\) buys more than \(1\) Euro (\(1.16\)), your Euro answer should always be a larger number than the pound amount!
Key Takeaway: Pounds to Foreign = Multiply. Foreign to Pounds = Divide.
Chapter Quick Review
• Money Notation: Always give answers to \(2\) decimal places (e.g., \(£12.40\)).
• Best Buys: Compare unit costs (e.g., price per \(1\text{ g}\) or per \(100\text{ g}\)).
• Net Pay: \(\text{Net Pay} = \text{Gross Pay} - \text{Deductions}\).
• Percentage Profit/Loss: \(\frac{\text{Profit or Loss}}{\text{Original Price}} \times 100\).
• Simple Interest: \(I = \frac{P \times R \times T}{100}\).
• Compound Interest: \(\text{Amount} = P \times (\text{Multiplier})^n\).
• Currency Exchange: Multiply to get foreign currency; divide to return to pounds.