Welcome to Financial Mathematics!
Have you ever wondered if a "Buy One Get One Half Price" deal is actually better than a "3 for 2" offer? Or how banks decide how much extra money to give you when you save with them? That is what Financial Mathematics is all about! In this chapter, we will learn how to use our math skills to make smart decisions with money in the real world. Because we are focusing on Working Mathematically, we won't just be doing sums; we will be learning how to solve problems and interpret results to become "money-smart."
1. The Basics: Working with Money
Before we dive into big problems, we need to be experts at the basics. In the UK, we use decimal notation for money. This means we use a decimal point to separate pounds (\( \text{£} \)) from pence (\( \text{p} \)).
Key Rules:
- \( 100\text{p} = \text{£}1.00 \)
- When using a calculator, the answer \( 4.5 \) in a money problem means \( \text{£}4.50 \). Never leave it as \( \text{£}4.5 \)!
- Always round money to 2 decimal places (the nearest penny) unless the problem asks for something else.
Quick Review: To change pence into pounds, we divide by 100. For example, \( 250\text{p} \div 100 = \text{£}2.50 \).
2. Unit Pricing (Finding the "Best Buy")
Have you ever been in a supermarket and seen two different sizes of the same cereal? Unit pricing helps us work out which one is better value by finding the cost of one single unit (like 1 gram, 100g, or 1 litre).
How to calculate Unit Price:
\( \text{Unit Price} = \frac{\text{Total Price}}{\text{Quantity}} \)
Example:
Shop A sells 2 litres of juice for \( \text{£}2.40 \).
Shop B sells 3 litres of juice for \( \text{£}3.30 \).
Which is better value?
Step 1: Shop A unit price \( = 2.40 \div 2 = \text{£}1.20 \) per litre.
Step 2: Shop B unit price \( = 3.30 \div 3 = \text{£}1.10 \) per litre.
Conclusion: Shop B is better value because it is cheaper for every litre you buy!
Key Takeaway: Don't just look at the total price. Calculate the cost per unit to see the real deal.
3. Percentages in Finance
Percentages are everywhere in money, from discounts (taking money off) to VAT/Tax (adding money on).
A. Percentage Increase and Decrease
To find a percentage of an amount, you can turn the percentage into a decimal (by dividing by 100) and multiply.
- To find \( 15\% \) of \( \text{£}60 \): \( 0.15 \times 60 = \text{£}9 \).
- Percentage Increase (e.g., Prices going up): If a \( \text{£}40 \) ticket increases by \( 10\% \), the increase is \( \text{£}4 \). The new price is \( 40 + 4 = \text{£}44 \).
- Percentage Decrease (e.g., A Sale): If a \( \text{£}100 \) coat has \( 20\% \) off, the discount is \( \text{£}20 \). The sale price is \( 100 - 20 = \text{£}80 \).
B. Finding the Original Value
Sometimes you know the price after a change and need to work backwards.
Example: A game costs \( \text{£}36 \) after a \( 10\% \) discount. What was the original price?
If it has \( 10\% \) off, then \( \text{£}36 \) represents \( 90\% \) of the original price.
\( 90\% = 36 \)
\( 1\% = 36 \div 90 = 0.4 \)
\( 100\% = 0.4 \times 100 = \text{£}40 \).
Key Takeaway: Always think about what percentage you have left after a discount!
4. Simple Interest
When you put money in a bank, the bank pays you interest as a reward for keeping your money there. Simple Interest means the interest is only calculated on the original amount you put in (the Principal).
The Simple Interest Method:
1. Find the percentage of the original amount.
2. Multiply that amount by the number of years.
Example:
You invest \( \text{£}500 \) at a Simple Interest rate of \( 3\% \) per year for 4 years.
Step 1: Find \( 3\% \) of \( \text{£}500 \).
\( 0.03 \times 500 = \text{£}15 \) (This is the interest you get every year).
Step 2: Multiply by the number of years.
\( 15 \times 4 = \text{£}60 \) (Total interest).
Step 3: If asked for the total balance, add it back to the start.
\( 500 + 60 = \text{£}560 \).
Don't worry if this seems tricky: Just remember that in "Simple Interest," the amount of money added stays the same every single year!
5. Solving Multi-Step Problems
In your exams, you will often have to solve "Working Mathematically" problems that have several steps. This is where you combine everything you know.
Problem-Solving Checklist:
1. Read carefully: Is it an increase or a decrease?
2. Identify units: Are some prices in pence and some in pounds? Convert them so they match!
3. Check your results: Does your answer make sense? If you buy a shirt in a sale, the price should be lower than the start, not higher!
4. Show your working: Write down each step so your teacher can see your brilliant logic.
Did you know? Banks used to calculate all interest by hand before computers! This is why "Simple Interest" was very popular—it was much faster to calculate than complex interest types.
Summary: Key Takeaways
- Money is Decimal: Always use two decimal places for pence (e.g., \( \text{£}5.70 \)).
- Best Buys: Use division to find the cost per unit to compare different shops.
- Percentages: Use decimals to find percentages quickly (e.g., \( 20\% \) is \( 0.2 \)).
- Simple Interest: Calculate the interest for one year and then multiply by the number of years.
- Logic: Always ask yourself, "Does this answer make sense in a real shop?"
For more on how to check if your answers are reasonable, see the chapter on Estimation and Checking Results.