Introduction to Percentages and Financial Maths

Percentages are everywhere! Whether you are looking at a sale in a shop, checking the battery life on your phone, or understanding how much interest you will earn on your savings, percentages help us compare quantities easily. In this chapter, we will look at how to master percentages and apply them to real-world money situations. Don't worry if you find this tricky at first—once you learn the multiplier method, you will find these calculations much simpler!

1. The Basics: What is a Percentage?

The word "percent" literally means "per hundred". Think of it like a hundred-square grid: \(15\%\) just means 15 out of every 100 squares are shaded.

To succeed in your GCSE, you need to be able to switch between fractions, decimals, and percentages (FDP) effortlessly. This is because decimals are often the easiest way to do calculations on a calculator.

The Multiplier Method

A multiplier is a decimal used to represent a percentage. To find the multiplier, simply divide the percentage by 100.

Example:
To find \(12\%\) of something, the multiplier is \(12 \div 100 = 0.12\)
To find \(7\%\) of something, the multiplier is \(7 \div 100 = 0.07\)

Key Takeaway: Always think of percentages as "parts per hundred." To get a decimal multiplier, divide by 100.

2. Calculating Percentage Change

In your exam, you might be asked to increase or decrease a value by a percentage. You can do this in one step using a multiplier.

Percentage Increase

Think of the original amount as \(100\%\). If you increase it by \(15\%\), you now have \(115\%\) of the original amount.
Multiplier: \(1.15\)

Example: Increase \(\$60\) by \(20\%\).
\(100\% + 20\% = 120\%\)
Multiplier = \(1.2\)
Calculation: \(60 \times 1.2 = 72\)

Percentage Decrease

If you decrease an amount by \(15\%\), you have \(85\%\) left (\(100 - 15 = 85\)).
Multiplier: \(0.85\)

Example: A \(\$200\) coat is in a \(30\%\) off sale.
\(100\% - 30\% = 70\%\)
Multiplier = \(0.7\)
Calculation: \(200 \times 0.7 = 140\)

Common Mistake to Avoid: When decreasing by \(5\%\), the multiplier is \(0.95\), not \(0.5\)! Be careful with those single-digit percentages.

3. Expressing One Quantity as a Percentage of Another

Sometimes you need to find out what the percentage actually is. For example, "What is \(35\) as a percentage of \(50\)?"

The Formula:
\(\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\)

Example: You score 18 out of 25 in a test.
\(\frac{18}{25} \times 100 = 72\%\)

Finding Percentage Profit or Loss

This is a common "real-world" question. Use this specific version of the formula:
\(\frac{\text{Change (Profit or Loss)}}{\text{Original Amount}} \times 100\)

Example: You buy a game for \(\$40\) and sell it for \(\$50\).
Profit = \(50 - 40 = 10\)
\(\frac{10}{40} \times 100 = 25\%\) profit.

4. Reverse Percentages (Original Value Problems)

These are "working backwards" problems. The question will give you the final price after an increase or decrease and ask for the original price.

The Golden Rule: Never calculate the percentage of the new price. You must divide by the multiplier.

Step-by-Step:
1. Find the multiplier (e.g., a \(20\%\) increase is \(1.2\)).
2. Divide the New Value by the Multiplier to find the Original Value.

Example: A car is sold for \(\$9600\) after a \(20\%\) price drop. What was the original price?
1. Price drop of \(20\%\) means we have \(80\%\) left. Multiplier = \(0.8\).
2. \(\$9600 \div 0.8 = \$12,000\)

5. Financial Maths: Interest and Taxes

The Edexcel syllabus expects you to understand household finance contexts. Here are the key terms:

  • Credit: Money going into an account.
  • Debit: Money going out of an account.
  • Balance: How much is in the account right now.
  • VAT: Value Added Tax (a percentage added to the cost of goods).
  • Income Tax: A percentage of your earnings paid to the government.

Simple Interest

Simple interest is calculated only on the original amount. You get the same amount of interest every year.

Example: Invest \(\$500\) for 3 years at \(2\%\) simple interest per year.
\(2\%\) of \(\$500 = 10\).
In 3 years, you get \(10 \times 3 = 30\).
Total amount = \(500 + 30 = 530\).

Compound Interest (Growth)

Compound interest is where you earn "interest on your interest." This is a growth problem. You will find this formula on your Exam Aid sheet:

\(\text{Total Accrued} = P (1 + \frac{r}{100})^n\)

Where:
\(P\) = Principal (starting amount)
\(r\) = Rate of interest
\(n\) = Number of years

Example: Invest \(\$2000\) for 4 years at \(3\%\) compound interest.
Calculation: \(2000 \times (1.03)^4\)
\(2000 \times 1.1255... = \$2251.02\) (rounded to 2 decimal places for money).

Did you know? Compound interest grows much faster than simple interest over long periods of time. This is why saving early is so powerful!

Summary Checklist

Before the exam, make sure you can:
1. Convert any percentage to a decimal multiplier.
2. Calculate percentage increase and decrease in one step.
3. Find the percentage profit or loss using the "change over original" formula.
4. Solve "reverse percentage" problems by dividing by the multiplier.
5. Use the compound interest formula from the formula sheet correctly.

Note: For more advanced topics like "Iterative processes" or "Depreciation," check the Growth, Decay and Iterative Processes chapter.