Financial Mathematics: Interest and Currency

TL;DR

This topic helps you understand how money changes value through profit/loss, discounts, and currency exchange. You'll learn to calculate simple and compound interest, which are crucial for loans and investments. Finally, we'll cover ratios and sequences, useful tools for spotting financial trends.

1. The Mental Model

Think of financial mathematics as the toolkit for understanding how money moves and grows (or shrinks!). It helps you make sense of prices, savings, loans, and even international spending.

2. The Core Material

Percentage Profit and Loss, Discount

When you buy or sell something, the price often changes.
* Profit: If you sell something for more than you bought it, you make a profit. It's usually expressed as a percentage of the original cost.
* Profit = Selling Price - Cost Price
* Percentage Profit = (Profit / Cost Price) * 100%
* Loss: If you sell something for less than you bought it, you make a loss.
* Loss = Cost Price - Selling Price
* Percentage Loss = (Loss / Cost Price) * 100%
* Discount: This is a reduction in the original price.
* Discount Amount = Original Price * (Discount Percentage / 100)
* New Price = Original Price - Discount Amount

Example: A shirt costs $40. You sell it for $50.
Profit = $50 - $40 = $10.
Percentage Profit = ($10 / $40) * 100% = 25%.

Example: A $60 item has a 10% discount.
Discount Amount = $60 * (10 / 100) = $6.
New Price = $60 - $6 = $54.

Currency Conversion

When you travel or buy things online from another country, you need to convert money. An exchange rate tells you how much one currency is worth in another.

• To convert from Currency A to Currency B: Multiply the amount in Currency A by the exchange rate (if the rate is "1 unit of A = X units of B").
• To convert from Currency B to Currency A: Divide the amount in Currency B by the exchange rate.

Example: 1 USD = 0.90 EUR.
To convert $100 USD to EUR: 100 * 0.90 = 90 EUR.
To convert 180 EUR to USD: 180 / 0.90 = 200 USD.

Simple and Compound Interest

Interest is the cost of borrowing money or the reward for saving it.

• Simple Interest: Interest is calculated only on the original principal amount. It's the same amount each period.

  • Interest (I) = Principal (P) * Rate (R) * Time (T)
  • Total Amount (A) = P + I
  • R is usually an annual rate, so T must be in years.

• Compound Interest: Interest is calculated on the principal amount AND on the accumulated interest from previous periods. This means your money grows faster.

  • Total Amount (A) = P * (1 + R)^T
  • R is the annual interest rate expressed as a decimal (e.g., 5% is 0.05).
  • T is the number of compounding periods (usually years).

Example (Simple Interest): You invest $1000 at 5% simple interest for 3 years.
I = 1000 * 0.05 * 3 = $150.
A = 1000 + 150 = $1150.

Example (Compound Interest): You invest $1000 at 5% compound interest for 3 years.
A = 1000 * (1 + 0.05)^3
A = 1000 * (1.05)^3
A = 1000 * 1.157625 = $1157.63.
Notice how compound interest gives you more!

graph TD
    Start["Start with Principal (P)"] --> Rate["Apply Interest Rate (R)"]
    Rate --> Time["For a Period (T)"]
    Time --> CalculationType{{"Which type of interest?"}}
    CalculationType --> Simple["Simple Interest"]
    Simple --> SimpleFormula["Interest = P * R * T"]
    SimpleFormula --> SimpleTotal["Total = P + Interest"]
    CalculationType --> Compound["Compound Interest"]
    Compound --> CompoundFormula["Total = P * (1 + R)^T"]
    CompoundFormula --> End["End with Total Amount"]

Ratios

A ratio compares two or more quantities. For example, if you mix juice, a 1:2 ratio of concentrate to water means for every 1 part concentrate, you add 2 parts water. In finance, ratios can compare things like profit to sales, or debt to equity.

Example: The ratio of profit to sales for a company is 3:10. If sales are $10,000, what's the profit?
For every $10 in sales, there's $3 in profit.
Sales are 10,000 / 10 = 1,000 times bigger than the ratio part.
So, Profit = 3 * 1,000 = $3,000.

Sequences

A sequence is a list of numbers that follow a specific pattern.

• Finding the nth term: This means finding a general rule or formula that lets you calculate any term in the sequence if you know its position (n).
  • Arithmetic sequences: Each term is found by adding a constant difference to the previous term.
    • nth term = a + (n-1)d, where a is the first term and d is the common difference.
  • Geometric sequences: Each term is found by multiplying the previous term by a constant ratio.
    • nth term = a * r^(n-1), where a is the first term and r is the common ratio.

Example (Arithmetic): Sequence: 2, 5, 8, 11...
Here, a = 2 and d = 3.
The nth term = 2 + (n-1)3.
For n=4, 2 + (4-1)3 = 2 + 9 = 11. Correct.

Example (Geometric): Sequence: 3, 6, 12, 24...
Here, a = 3 and r = 2.
The nth term = 3 * 2^(n-1).
For n=3, 3 * 2^(3-1) = 3 * 2^2 = 3 * 4 = 12. Correct.

• Justifying/Proving General Rules: This involves showing why a formula works using logical steps or algebra. For example, for an arithmetic sequence, you can show that a_n = a_1 + (n-1)d by demonstrating that the difference between a_n and a_{n-1} is always d.

Variation: Direct and Inverse

This describes how two quantities relate to each other.

• Direct Variation: As one quantity increases, the other increases proportionally.
  • y = kx (where k is the constant of proportionality).
  • Example: The more hours you work, the more money you earn (if your wage is constant).
• Inverse Variation: As one quantity increases, the other decreases proportionally.
  • y = k/x (where k is the constant of proportionality).
  • Example: The more people you have doing a job, the less time it takes.

Example (Direct): If y varies directly with x, and y=10 when x=2, find y when x=5.
First, find k: 10 = k * 2 => k = 5.
Then, y = 5 * 5 = 25.

Example (Inverse): If y varies inversely with x, and y=6 when x=2, find y when x=4.
First, find k: 6 = k / 2 => k = 12.
Then, y = 12 / 4 = 3.

3. Worked Example

Let's combine a few concepts. You're starting a small online business. You buy an item for £80. You want to make a 20% profit. You also need to consider currency exchange. A customer in the US wants to buy it, and the exchange rate is £1 = $1.25.

1. Calculate desired selling price in GBP (£) with profit:

   • Profit amount = 20% of £80 = 0.20 * 80 = £16.
   • Selling price in GBP = £80 (cost) + £16 (profit) = £96.

2. Convert selling price to USD ($):

   • Selling price in USD = £96 * $1.25/£ = $120.

So, you'd list the item for $120 to achieve your 20% profit after currency conversion.

4. Key Takeaways

• Profit and loss are calculated as a percentage of the original cost, while discounts are a percentage of the original price.
• Currency exchange rates define how much one currency is worth compared to another.
• Simple interest is fixed on the principal, while compound interest grows on both the principal and accumulated interest, making your money grow faster.
• Ratios help compare quantities, often simplifying complex financial relationships.
• Sequences identify patterns in numbers, which can be useful for predicting financial trends.
• Direct variation means two quantities increase or decrease together proportionally, while inverse variation means one increases as the other decreases.

Common Mistakes to Avoid:

• Confusing the base for percentage calculations (e.g., calculating profit percentage on selling price instead of cost price).
• Mixing up simple and compound interest formulas, leading to incorrect calculations for growth.
• Dividing instead of multiplying (or vice-versa) during currency conversions.
• Forgetting to convert the interest rate to a decimal or time to years when using interest formulas.
• Misidentifying if a sequence is arithmetic or geometric before finding its nth term.

5. Now Try It

You want to buy a new laptop that costs $1200. The store offers a 15% discount. You also have a savings account that offers 3% compound interest annually. If you saved $1000 three years ago in that account, do you have enough money to buy the discounted laptop now? If not, how much more do you need?

Success looks like:
1. Calculating the discounted price of the laptop.
2. Calculating the total amount of money in your savings account after 3 years.
3. Comparing these two amounts and stating the difference if you can't afford it.