Welcome to the World of Percentages!
Hello! Get ready to explore the super useful topic of percentages. You see them everywhere: on sale signs at your favourite shop, on your test results, in news about the economy, and even on food labels. Understanding them is like having a superpower for making smart decisions in everyday life!
In these notes, we'll break everything down into easy steps. We will learn:
- How to calculate percentage changes (like when a price goes up or down).
- How to solve real-life problems involving discounts, profit and loss, growth and depreciation, interest, and taxation (rates, property tax, and salaries tax).
Don't worry if this seems tricky at first. We'll go through it together with lots of examples. Let's get started!
Section 1: The Heart of the Matter - Percentage Change
Everything changes, right? Prices go up, things go on sale, your height increases! Percentage change is a way to measure exactly how much something has changed compared to its starting value.
Finding the Percentage Change
There is one master formula that works for both increases and decreases. It's a lifesaver!
The Master Formula:
\( \text{Percentage Change} = \frac{\text{Change in Value}}{\text{Original Value}} \times 100\% \)
Let's break that down:
- Original Value: This is the starting amount. It's what you had before the change. This is the most important number!
- Change in Value: This is the difference between the new value and the original value. Just subtract! (New Value - Original Value).
Example 1: Percentage Increase (Growth)
Imagine your favourite bubble tea shop increases the price of your usual drink from \$20 to \$25. What is the percentage increase?
Step-by-step:
1. Find the Original Value: The price was \$20 at the start. So, Original Value = \$20.
2. Find the Change in Value: The price went up by \$25 - \$20 = \$5. So, Change = \$5.
3. Use the Master Formula:
\( \frac{\$5}{\$20} \times 100\% = 0.25 \times 100\% = 25\% \)
Answer: The price increased by 25%.
Example 2: Percentage Decrease
You've been saving up for a video game that was \$400. This week, it's on sale for \$320! What is the percentage decrease?
Step-by-step:
1. Find the Original Value: The game's original price was \$400. So, Original Value = \$400.
2. Find the Change in Value: The price dropped by \$400 - \$320 = \$80. So, Change = \$80.
3. Use the Master Formula:
\( \frac{\$80}{\$400} \times 100\% = 0.2 \times 100\% = 20\% \)
Answer: The price decreased by 20%.
Key Takeaway for Percentage Change
To find a percentage change, always remember: Divide the actual change by the original amount, then multiply by 100%. The "original" part is the key!
Section 2: Percentages in Real Life - Solving Problems
Now let's use our percentage skills to tackle some common situations. This is where maths becomes really powerful!
2.1 Discount
Everyone loves a sale! A discount is a percentage decrease on an item's original price.
Key Terms:
- Marked Price: The original price on the tag.
- Discount: The percentage you get "off" (e.g., 30% off).
- Sale Price: The final price you actually pay.
Example: Finding the Sale Price
A pair of headphones has a marked price of \$500. There is a "20% off" discount. What is the sale price?
Method 1: Find the discount amount first.
1. Calculate the discount amount: \$500 \times 20\% = 500 \times 0.20 = \$100. This is how much money you save.
2. Subtract the discount from the marked price: \$500 - \$100 = \$400.
Method 2: A quicker way!
1. Think about what percentage you PAY: If you get 20% off, you still have to pay 100% - 20% = 80% of the price.
2. Calculate the sale price directly: \$500 \times 80\% = 500 \times 0.80 = \$400.
Both methods give the same answer: The sale price is \$400.
2.2 Profit and Loss
When businesses sell things, they either make a profit (make money) or a loss (lose money).
Key Terms:
- Cost Price (CP): How much it costs the business to buy an item.
- Selling Price (SP): How much the business sells the item for.
- Profit: If SP is greater than CP (Profit = SP - CP).
- Loss: If CP is greater than SP (Loss = CP - SP).
Important Note: Profit or loss percentage is always calculated based on the Cost Price!
The Formulas:
\( \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\% \)
\( \text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100\% \)
Example: Making a Profit
A shop owner buys a school bag for \$150 (Cost Price) and sells it for \$180 (Selling Price). What is her profit percentage?
1. Find the profit amount: \$180 (SP) - \$150 (CP) = \$30.
2. Calculate the profit percentage (using the CP!):
\( \frac{\$30}{\$150} \times 100\% = 0.2 \times 100\% = 20\% \)
Answer: The shop owner made a 20% profit.
2.3 Growth and Depreciation
These terms describe percentage changes repeated over time.
- Growth: When a quantity increases at a constant percentage rate \(r\) per period for \(n\) periods.
\( A = P(1 + r)^n \)
- Depreciation / Decay: When the value of an item decreases at a constant percentage rate \(r\) per period for \(n\) periods.
\( A = P(1 - r)^n \)
Example: Multi-period Depreciation
A brand new car costs \$200,000. Its value depreciates at a rate of 15% per year. What is its value after 3 years?
Here, \(P = \$200,000\), \(r = 15\% = 0.15\), and \(n = 3\).
\( A = 200000 \times (1 - 0.15)^3 = 200000 \times (0.85)^3 = \$122,825 \)
Answer: The car is worth \$122,825 after 3 years.
2.4 Simple and Compound Interest
When you put money in a bank or take out a loan, interest is involved.
Simple Interest
With simple interest, you only earn interest on the original principal \(P\).
Formula: \( I = P \times R \times T \)
- I = The interest amount (\$)
- P = The Principal (starting money)
- R = The interest Rate per year
- T = The Time (in years)
Example: You deposit \$1000 into a bank account with a simple interest rate of 3% per annum for 5 years.
\( I = \$1000 \times 3\% \times 5 = \$150 \)
Total amount \(A = P + I = \$1000 + \$150 = \$1150\).
Compound Interest
With compound interest, interest earned in each period is added to the principal for calculating future interest.
General Formula:
\( A = P\left(1 + \frac{R}{k}\right)^{n \times k} \)
- A = Final Amount (Principal + Interest)
- P = Principal
- R = Annual interest rate
- n = Number of years
- k = Number of compounding periods per year (Annually: \(k=1\), Half-yearly: \(k=2\), Quarterly: \(k=4\), Monthly: \(k=12\))
- Compound Interest (CI): \( \text{CI} = A - P \)
Example: You deposit \$10,000 at 6% p.a. compounded quarterly for 2 years. Find the final amount and compound interest.
- Rate per quarter = \( \frac{6\%}{4} = 1.5\% = 0.015 \)
- Total compounding periods = \( 2 \times 4 = 8 \)
\( A = 10000 \times (1 + 0.015)^8 \approx \$11,264.93 \)
\( \text{Interest} = \$11,264.93 - \$10,000 = \$1,264.93 \)
2.5 Successive Percentage Changes
When multiple percentage changes happen in succession, calculate each step using the updated base value!
Example: Successive Discounts
An item costs \$100. It gets a 10% discount, followed by another 10% discount.
1. After 1st discount: \( \$100 \times (1 - 10\%) = \$90 \)
2. After 2nd discount: \( \$90 \times (1 - 10\%) = \$81 \)
Overall Percentage Change: \( \frac{\$81 - \$100}{\$100} \times 100\% = -19\% \) (a 19% total discount, not 20%!).
2.6 Taxation and Rates
Governments collect taxes and rates to fund public services and infrastructure.
A. Rates
Rates are a tax levied on properties, calculated as a percentage of the Rateable Value (the estimated annual rental value).
\( \text{Rates payable for a year} = \text{Rateable Value} \times \text{Rates Percentage} \)
Rates are usually paid quarterly (every 3 months): \( \text{Quarterly Rates} = \frac{\text{Annual Rates}}{4} \).
Example: A flat has a rateable value of \$120,000. If the rates percentage is 5% per annum, find the quarterly rates payable.
\( \text{Annual Rates} = \$120,000 \times 5\% = \$6,000 \)
\( \text{Quarterly Rates} = \frac{\$6,000}{4} = \$1,500 \)
B. Property Tax
Property tax is charged to property owners based on the rental income from leasing a property.
1. Net Assessable Value (NAV): \( \text{NAV} = \text{Annual Rental Income} \times (1 - 20\%) \) (where 20% is the statutory allowance for repairs and outgoings).
2. Property Tax: \( \text{Property Tax} = \text{Net Assessable Value} \times \text{Property Tax Rate} \)
Example: An owner rents out a shop for \$30,000 per month. If the property tax rate is 15%, find the property tax payable for the year.
- Annual Rental Income = \( \$30,000 \times 12 = \$360,000 \)
- Net Assessable Value = \( \$360,000 \times (1 - 20\%) = \$288,000 \)
- Property Tax = \( \$288,000 \times 15\% = \$43,200 \)
C. Salaries Tax
Salaries tax is calculated on income after subtracting allowances, using progressive tax rates (or a standard rate).
\( \text{Net Chargeable Income (NCI)} = \text{Total Income} - \text{Allowances} - \text{Deductions} \)
Example: Suppose the progressive tax rates are:
- First \$50,000 of NCI at 2%
- Next \$50,000 of NCI at 6%
- Remaining NCI at 10%
If a person has a Net Chargeable Income of \$130,000, calculate their tax payable.
1. First \$50,000: \( \$50,000 \times 2\% = \$1,000 \)
2. Next \$50,000: \( \$50,000 \times 6\% = \$3,000 \)
3. Remaining \$30,000: \( (\$130,000 - \$100,000) \times 10\% = \$3,000 \)
4. Total Salaries Tax: \( \$1,000 + \$3,000 + \$3,000 = \$7,000 \)
Chapter Summary: Quick Review
Here are the key formulas and concepts for Percentages:
- Percentage Change: \( \frac{\text{Change}}{\text{Original}} \times 100\% \)
- Profit / Loss %: Calculated relative to the Cost Price.
- Growth & Depreciation: \( A = P(1 + r)^n \) or \( A = P(1 - r)^n \)
- Simple Interest: \( I = P \times R \times T \)
- Compound Interest: \( A = P\left(1 + \frac{R}{k}\right)^{n \times k} \)
- Successive Changes: Apply changes one by one to the updated value.
- Taxation & Rates: Understand Rates (Rateable Value), Property Tax (Net Assessable Value), and Salaries Tax (Net Chargeable Income).
Keep practising with these ideas, and soon you'll be a percentages expert. You've got this!