Welcome to the World of Percentages!

Hey there! Ready to unlock one of the most useful skills in mathematics? This chapter is all about percentages. You see them everywhere: on sale signs at your favourite shop, on your phone's battery indicator, and in news reports. Understanding them is like having a secret code for how the world works with numbers.

In these notes, we'll break down everything you need to know. We'll learn how to calculate when things increase or decrease, figure out discounts, understand profit and loss, and even see how money can grow with interest. Don't worry if it sounds like a lot – we'll take it one step at a time with plenty of real-life examples. Let's get started!


Section 1: Understanding Percentage Change

The most important idea in this chapter is percentage change. It's just a way to measure how much a number has changed compared to its original value. It can either go up (an increase) or go down (a decrease).

The Master Formula for Percentage Change

Almost everything we do in this chapter comes back to one simple idea. To find the percentage change, you use this formula:

$$ \text{Percentage Change} = \frac{\text{Change in Value}}{\text{Original Value}} \times 100\% $$

Remember: The "Change in Value" is just the difference between the new value and the original value.

Percentage Increase (When things get bigger!)

This happens when the new value is more than the original value.

Example: Last month, your allowance was $200. This month, it's $220. What is the percentage increase?

  1. Find the Change: $220 - $200 = $20 increase.
  2. Use the Formula: $$ \frac{\text{Increase}}{\text{Original Amount}} \times 100\% = \frac{$20}{$200} \times 100\% $$
  3. Calculate: $0.1 \times 100\% = 10\%$.

So, your allowance increased by 10%. Well done!

Percentage Decrease (When things get smaller!)

This happens when the new value is less than the original value.

Example: A video game originally cost $400. It's now on sale for $300. What is the percentage decrease?

  1. Find the Change: $400 - $300 = $100 decrease.
  2. Use the Formula: $$ \frac{\text{Decrease}}{\text{Original Amount}} \times 100\% = \frac{$100}{$400} \times 100\% $$
  3. Calculate: $0.25 \times 100\% = 25\%$.

The price of the game decreased by 25%. What a bargain!


Key Takeaway: Percentage Change

To calculate any percentage change, remember these two steps:

  • Step 1: Find the actual change (New Value - Original Value).
  • Step 2: Divide the change by the original value and multiply by 100%.

Section 2: Percentages in Real Life

Now let's see how these ideas work in everyday situations. This is where maths becomes super useful!

Topic 1: Discount and Sales

You love sales, right? A discount is just a percentage decrease applied to a price.

Key Terms:

  • Marked Price: The original price tag on the item.
  • Discount %: The percentage you get off.
  • Selling Price: The final price you actually pay.

How to calculate the Selling Price:

Example: A T-shirt has a marked price of $150 and is on sale with a 20% discount. What is the selling price?

Method 1: Find the discount first.

  1. Calculate the discount amount: $150 \times 20\% = 150 \times 0.20 = $30.
  2. Subtract it from the marked price: $150 - $30 = $120.

Method 2: Use a multiplier (Faster!)

If you get 20% off, you are still paying 80% of the price (since 100% - 20% = 80%).

  1. Calculate the final percentage you pay: 100% - 20% = 80%.
  2. Multiply the marked price by this percentage: $150 \times 80\% = 150 \times 0.80 = $120.

The selling price is $120.


Topic 2: Profit and Loss

Businesses use percentages to see if they are making or losing money.

Key Terms:

  • Cost Price: How much it costs a shop to buy an item.
  • Selling Price: How much the shop sells the item for.
  • Profit: If Selling Price > Cost Price. You made money!
  • Loss: If Cost Price > Selling Price. You lost money.
Very Important Rule!

Profit or Loss Percentage is ALWAYS calculated based on the COST PRICE. Always go back to what it originally cost!

Example (Profit): A shop owner buys a watch for $500 (Cost Price) and sells it for $600 (Selling Price). Find her profit percentage.

  1. Find the profit amount: $600 - $500 = $100 profit.
  2. Use the formula (remember to use Cost Price!): $$ \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\% = \frac{$100}{$500} \times 100\% $$
  3. Calculate: $0.20 \times 100\% = 20\%$.

The shop owner made a 20% profit.


Topic 3: Growth and Depreciation

This is just a fancy way of saying "percentage increase or decrease over time".

  • Growth: An increase in value. Example: The population of a city.
  • Depreciation: A decrease in value. Example: The value of a new car or phone.

Example (Depreciation): A new phone costs $6000. Its value depreciates by 30% after one year. What is its value after one year?

This is just a percentage decrease problem! We can use the multiplier method:

  1. Find the remaining percentage value: 100% - 30% = 70%.
  2. Calculate the new value: $6000 \times 70\% = 6000 \times 0.70 = $4200.

The phone is worth $4200 after one year.


Topic 4: Interest - Making Money Grow!

When you put money in a bank, it earns interest. It's like a "thank you" fee from the bank for letting them use your money.

Key Terms:

  • Principal (P): The initial amount of money you invest.
  • Rate (R): The percentage of interest per year (p.a. = per annum = per year).
  • Time (T): The number of years the money is invested.
  • Interest (I): The extra money you earn.
  • Amount (A): The total money you have at the end (Principal + Interest).
Simple Interest

With simple interest, the interest is calculated only on the original Principal every year. It's the same amount of extra money each year.

Formula: $$ I = P \times R \times T $$

Example: You invest $1000 (P) at a simple interest rate of 5% p.a. (R) for 3 years (T). How much interest do you earn?

  1. Use the formula: $$ I = $1000 \times 5\% \times 3 $$
  2. Calculate: $$ I = 1000 \times 0.05 \times 3 = $150 $$

You earn $150 in simple interest. The total amount you have is $1000 + $150 = $1150.

Compound Interest

This is more powerful! With compound interest, you earn interest on your principal AND on the interest you've already earned. It's like a snowball rolling downhill, getting bigger and bigger.

Don't worry if this seems tricky at first, it's just a repeated percentage increase!

Formula for the total Amount (A): $$ A = P \times (1 + R)^T $$

Example: You invest $1000 (P) at a compound interest rate of 5% p.a. (R) for 3 years (T). How much will you have in total?

  1. Use the formula: $$ A = $1000 \times (1 + 5\%)^3 $$
  2. Calculate the part in the bracket first: $$ 1 + 0.05 = 1.05 $$
  3. Now the full calculation: $$ A = 1000 \times (1.05)^3 = 1000 \times 1.157625 = $1157.63 $$

You will have a total of $1157.63. That's $7.63 more than with simple interest! It might not seem like much now, but over many years, compounding makes a huge difference.


Did You Know?

The percent sign (%) evolved from a symbol used in the 15th century. It was originally written as "p c" with a small circle, which eventually became the two circles we see today separated by a line!


Topic 5: Successive and Component Changes

Successive Changes (One after another)

This is when you apply one percentage change, and then another percentage change to the new result.

Example: A jacket costs $500. It is first discounted by 20%. Then, for a special weekend, there is a further 10% discount. What is the final price?

Common Mistake Alert!

You CANNOT just add the percentages (20% + 10% = 30%). This is wrong because the second discount is calculated on the already reduced price, not the original price.

The Right Way (Step-by-step):

  1. Price after the first 20% discount: $500 \times (100\% - 20\%) = 500 \times 0.80 = $400.
  2. Now, apply the 10% discount to this new price: $400 \times (100\% - 10\%) = 400 \times 0.90 = $360.

The final price is $360. (A 30% discount would have been $350, so you can see it's different!)

Component Changes (Changes in different parts of a whole)

This is when different parts of a total change by different percentages.

Example: A school has 40% boys and 60% girls. Next year, the number of boys will increase by 10%, and the number of girls will decrease by 5%. What is the overall percentage change in the school's total population?

The Easiest Way: Assume a starting total. Let's say the school has 100 students.

  1. Original numbers: 40 boys and 60 girls. Total = 100.
  2. New number of boys: An increase of 10%. So, $40 \times (1 + 10\%) = 40 \times 1.1 = 44$ boys.
  3. New number of girls: A decrease of 5%. So, $60 \times (1 - 5\%) = 60 \times 0.95 = 57$ girls.
  4. New total population: $44 + 57 = 101$ students.
  5. Calculate the overall percentage change: The population went from 100 to 101. That's a 1% increase overall.

Topic 6: Salaries Tax

This is a real-world tax that adults pay on the money they earn (their salary).

Key Terms:

  • Income: The total amount of money a person earns.
  • Allowance: An amount of money you can subtract from your income that is not taxed.
  • Net Chargeable Income: The amount of income that tax is actually calculated on.
  • Tax Rate: The percentage used to calculate the tax.
  • Tax Payable: The final amount of tax you must pay.

The Process:

  1. Find the Net Chargeable Income: $$ \text{Net Chargeable Income} = \text{Income} - \text{Allowances} $$
  2. Calculate the Tax Payable: $$ \text{Tax Payable} = \text{Net Chargeable Income} \times \text{Tax Rate} $$

Example: Mr. Chan earns $400,000 a year. His total allowances are $150,000. Let's say the tax rate is 15%. How much tax does he pay?

  1. Find his Net Chargeable Income: $400,000 - $150,000 = $250,000.
  2. Calculate the Tax Payable: $250,000 \times 15\% = 250,000 \times 0.15 = $37,500.

Mr. Chan's tax payable is $37,500 for the year.

(Note: In real life, tax systems are often more complicated with different rates for different levels of income, but the basic idea is the same!)


Final Key Takeaway

Congratulations! You've covered all the key uses of percentages. From shopping to banking, you now have the tools to understand how these numbers work. The most important thing is to read the question carefully, identify the original value (like the cost price or the marked price), and then apply the correct percentage change. You've got this!