Your Awesome Guide to Using Percentages!
Hey there! Ready to unlock the power of percentages? It might sound like just another maths topic, but percentages are everywhere in real life. From figuring out a discount at your favourite shop, to understanding your phone's battery life, or even calculating interest on savings – percentages are super useful!
In these notes, we'll break everything down into easy-to-understand steps. We'll look at how things change (like prices going up or down) and solve real-world problems. Don't worry if you find maths tricky, we'll go through it together. You've got this!
1. What is a Percentage? (A Quick Refresher)
The word "percent" literally means "per 100". So, a percentage is just a special fraction where the bottom number (the denominator) is always 100.
Think of a giant pizza cut into 100 slices. If you eat 25 slices, you've eaten 25 out of 100, which is 25% of the pizza!
Converting Between Forms
To be a percentage pro, you need to be able to switch between percentages, decimals, and fractions. Here's a quick reminder:
- Percent to Decimal: Divide by 100 (or just move the decimal point two places to the left).
Example: 75% becomes 0.75 - Decimal to Percent: Multiply by 100 (or just move the decimal point two places to the right).
Example: 0.4 becomes 40% - Percent to Fraction: Put the number over 100 and simplify if you can.
Example: 50% becomes $$ \frac{50}{100} $$ which simplifies to $$ \frac{1}{2} $$
2. Percentage Change: The Up and Down of Numbers
This is the most important idea in this chapter! Percentage change tells us how much a quantity has increased or decreased compared to its original amount.
Percentage Increase
This is when a value gets bigger. Think about the price of a game increasing, or you growing taller!
The Formula:
$$ \text{Percentage Increase} = \frac{\text{The Amount of Increase}}{\text{Original Amount}} \times 100\% $$Step-by-step Example:
Last year, a concert ticket cost $500. This year, it costs $550. What is the percentage increase in price?
- Find the Increase: How much did the price go up?
$$ $550 - $500 = $50 $$ - Use the Formula: Divide the increase by the original price.
$$ \frac{\text{Increase}}{\text{Original Amount}} = \frac{$50}{$500} = 0.1 $$ - Convert to a Percentage: Multiply your answer by 100%.
$$ 0.1 \times 100\% = 10\% $$
Answer: The price of the ticket increased by 10%.
Percentage Decrease
This is when a value gets smaller. This happens when something is on sale, or when your phone battery goes down.
The Formula:
$$ \text{Percentage Decrease} = \frac{\text{The Amount of Decrease}}{\text{Original Amount}} \times 100\% $$Step-by-step Example:
A t-shirt was originally $120, but it's on sale for $90. What is the percentage decrease?
- Find the Decrease: How much did the price go down?
$$ $120 - $90 = $30 $$ - Use the Formula: Divide the decrease by the original price.
$$ \frac{\text{Decrease}}{\text{Original Amount}} = \frac{$30}{$120} = 0.25 $$ - Convert to a Percentage: Multiply by 100%.
$$ 0.25 \times 100\% = 25\% $$
Answer: The price of the t-shirt decreased by 25%.
Quick Review: Percentage Change
The most important rule is to ALWAYS divide by the ORIGINAL amount. Whether it's an increase or decrease, the starting value is your base. A common mistake is to divide by the new amount – don't fall into that trap!
Key Takeaway: Percentage change is all about comparing the change to where it started.
3. Solving Real-Life Problems
Now let's use what we've learned to tackle some common real-world situations. This is where maths becomes a superpower!
3.1 Discount, Profit, and Loss
This is all about the world of shopping and business!
Discount
A discount is a type of percentage decrease. It's the amount of money taken off the original price.
Example: A pair of headphones costs $800. It is on sale with a 20% discount. What is the sale price?
Method 1: Find the discount amount first.
- Calculate the discount: $$ 20\% \text{ of } $800 = 0.20 \times $800 = $160 $$
- Subtract from the original price: $$ $800 - $160 = $640 $$
Method 2: The Quick Method!
- Find the percentage you pay: If there's a 20% discount, you still have to pay $$ 100\% - 20\% = 80\% $$ of the price.
- Calculate the final price: $$ 80\% \text{ of } $800 = 0.80 \times $800 = $640 $$
Both methods work! The quick method is faster once you get the hang of it.
Profit and Loss
Let's learn some business language:
- Cost Price (CP): The price a shop pays for an item.
- Selling Price (SP): The price the shop sells the item to you for.
- Profit: If the Selling Price is MORE than the Cost Price (SP > CP).
- Loss: If the Selling Price is LESS than the Cost Price (SP < CP).
Percentage Profit/Loss Formula:
$$ \text{Percentage Profit} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\% $$ $$ \text{Percentage Loss} = \frac{\text{Loss}}{\text{Cost Price}} \times 100\% $$Common Mistake Alert! Always, always, always calculate the percentage profit or loss using the Cost Price as the original amount. It's the shop's starting point!
Example: A shop buys a watch for $400 (the Cost Price) and sells it for $500 (the Selling Price). What is the percentage profit?
- Find the profit: $$ \text{Profit} = SP - CP = $500 - $400 = $100 $$
- Use the formula: $$ \text{Percentage Profit} = \frac{$100}{$400} \times 100\% = 0.25 \times 100\% = 25\% $$
Answer: The shop made a 25% profit.
3.2 Growth and Depreciation
Some things gain value over time (growth), while others lose value (depreciation).
- Examples of Growth: Population of a city, money in a savings account.
- Examples of Depreciation: The value of a new car or phone after a year.
Example (Growth): The population of a town is 10,000. It is expected to grow by 5% next year. What will the new population be?
This is just a 5% increase! We can use the quick method.
New Population = Original × (100% + Growth %)
$$ = 10,000 \times (100\% + 5\%) $$
$$ = 10,000 \times 105\% $$
$$ = 10,000 \times 1.05 = 10,500 $$
Example (Depreciation): You buy a new phone for $6,000. Its value depreciates by 30% in the first year. What is it worth after one year?
This is a 30% decrease.
New Value = Original × (100% - Depreciation %)
$$ = $6,000 \times (100\% - 30\%) $$
$$ = $6,000 \times 70\% $$
$$ = $6,000 \times 0.70 = $4,200 $$
3.3 Simple and Compound Interest
When you put money in a bank, it earns interest. There are two main types:
Simple Interest
You only earn interest on the original amount of money you put in (the Principal). It's the same amount of interest every year.
Key Terms:
- Principal (P): The starting amount of money.
- Interest Rate (R): The percentage at which the money grows per year. (Remember to use it as a decimal in the formula!)
- Time (T): The number of years the money is invested for.
Formula: $$ \text{Simple Interest (I)} = P \times R \times T $$
Example: You deposit $2,000 (P) into a bank account with a simple interest rate of 3% (R) per year for 4 years (T). How much interest will you earn?
- Convert R to a decimal: $$ 3\% = 0.03 $$
- Use the formula: $$ I = $2,000 \times 0.03 \times 4 = $240 $$
Answer: You will earn $240 in simple interest over 4 years.
Compound Interest
This is much more powerful! You earn interest on your principal AND on the interest you've already earned. Your money starts making its own money!
Did you know?
Albert Einstein reportedly called compound interest the "eighth wonder of the world." It's the secret to how savings can grow really big over a long time!
Formula for the Total Amount (A):
$$ A = P(1 + R)^n $$Where n is the number of times it's compounded (usually the number of years).
Example: Let's use the same numbers. You deposit $2,000 (P) at an interest rate of 3% (R) for 4 years (n), but this time it's compounded annually. What is the total amount in your account?
- Convert R to a decimal: $$ 3\% = 0.03 $$
- Use the formula: $$ A = $2,000 \times (1 + 0.03)^4 $$ $$ A = $2,000 \times (1.03)^4 $$ $$ A = $2,000 \times 1.1255... $$ $$ A \approx $2251.02 $$
Answer: The total amount will be $2251.02. The interest earned is $$ $2251.02 - $2000 = $251.02 $$, which is more than the $240 from simple interest!
3.4 Successive Percentage Changes
This is when you have more than one percentage change, one after the other. The key is that the second change is calculated on the new amount, not the original.
Example: A jacket costs $1,000. The price is first increased by 10%. Then, during a sale, the new price is discounted by 10%. Is the final price $1,000? Let's see!
- First Change (10% increase):
New Price = $$ $1,000 \times (100\% + 10\%) = $1,000 \times 1.10 = $1,100 $$ - Second Change (10% discount on the new price):
Final Price = $$ $1,100 \times (100\% - 10\%) = $1,100 \times 0.90 = $990 $$
Answer: The final price is $990, not $1,000! This is a classic trick question. A percentage increase and then the same percentage decrease will always result in a final value that is lower than the original.
3.5 Salaries Tax
When people earn money (a salary), they often have to pay a portion of it to the government as tax. This is often calculated as a percentage.
Example: Mr. Chan earns $40,000 a month. The salaries tax rate is 5%. How much tax does he have to pay?
This is a straightforward percentage calculation.
Tax Payable = Salary × Tax Rate
$$ = $40,000 \times 5\% $$
$$ = $40,000 \times 0.05 = $2,000 $$
Answer: Mr. Chan has to pay $2,000 in tax.
Chapter Summary & Master Formula Sheet
Wow, we've covered a lot! You can now handle almost any percentage problem life throws at you. Remember, the key is to read the question carefully and identify the 'original' amount.
Key Formulas to Remember:
- Percentage Change: $$ \frac{\text{Change}}{\text{Original Amount}} \times 100\% $$
- New Value (Quick Method): $$ \text{Original Value} \times (1 \pm \text{Percentage Change as decimal}) $$
- Percentage Profit/Loss: $$ \frac{\text{Profit or Loss}}{\text{Cost Price}} \times 100\% $$
- Simple Interest: $$ I = P \times R \times T $$
- Compound Interest Total: $$ A = P(1 + R)^n $$
Keep practising and you'll become a percentages master in no time. Great job working through these notes!