Welcome to the World of Percentage Change!
In this chapter, we are going to explore how numbers grow and shrink using percentages. Whether it’s a 20% discount on a new pair of trainers, a 10% increase in your character's health points in a video game, or tracking how much a plant has grown, percentage change is everywhere!
Don't worry if percentages have felt a bit "maths-heavy" before. By the end of these notes, you'll see that it’s just a way of comparing how much something has changed compared to where it started. Let’s dive in!
1. The Basics: What is a Percentage?
Before we change anything, let’s do a quick refresher. The word percent comes from the Latin "per centum," which literally means "out of 100."
Think of a percentage like a chocolate bar broken into 100 tiny pieces. If you have 50%, you have 50 out of those 100 pieces (half the bar!).
Quick Review: The Multiplier
To make calculations easier, we often turn percentages into decimals. We call this the multiplier. To get the multiplier, just divide the percentage by 100.
Example: 25% becomes \( 25 \div 100 = 0.25 \).
Example: 7% becomes \( 7 \div 100 = 0.07 \).
Key Takeaway: Always remember that a percentage is just a fraction of 100!
2. Percentage Increase
A percentage increase happens when we add a certain percentage of the original amount back onto itself. It’s like "levelling up."
How to do it (Step-by-Step):
There are two main ways to increase a number by a percentage. Let's look at an example: Increase £40 by 20%.
Method A: The "Find and Add" Method
1. Find 20% of £40: \( 0.20 \times 40 = 8 \).
2. Add that to the original: \( 40 + 8 = 48 \).
3. The answer is £48.
Method B: The Multiplier Method (Great for Calculators!)
1. Start with 100% (the original amount) and add the increase: \( 100\% + 20\% = 120\% \).
2. Turn 120% into a decimal: \( 1.2 \).
3. Multiply the original by this decimal: \( 40 \times 1.2 = 48 \).
4. The answer is £48.
Memory Trick: If you are increasing, your multiplier will always be more than 1 (e.g., 1.10, 1.50, 2.0). If you are decreasing, it will be less than 1!
Key Takeaway: To increase, you are essentially finding more than 100% of the original number.
3. Percentage Decrease
A percentage decrease is when a value goes down. This is what happens during a sale at a shop or when your phone battery runs low.
Example: A £60 hoodie is in a "30% off" sale. What is the new price?
Method A: The "Find and Subtract" Method
1. Find 30% of £60: \( 0.30 \times 60 = 18 \).
2. Subtract that from the original: \( 60 - 18 = 42 \).
3. The new price is £42.
Method B: The Multiplier Method
1. Start with 100% and subtract the decrease: \( 100\% - 30\% = 70\% \).
2. Turn 70% into a decimal: \( 0.7 \).
3. Multiply the original by this decimal: \( 60 \times 0.7 = 42 \).
4. The new price is £42.
Did you know? Using the multiplier method (Method B) is much faster when you are using a calculator, especially for tricky numbers like a 17.5% decrease!
Key Takeaway: Percentage decrease means you are left with less than 100% of what you started with.
4. Calculating the "Percentage Change"
Sometimes, we know the old price and the new price, but we want to work out what the percentage change was. For example: "My YouTube channel had 200 subscribers last month, and now it has 250. What was the percentage increase?"
To find this, we use the Golden Formula:
\( \text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100 \)
Step-by-Step Example:
A plant was 10cm tall. After a week, it is 15cm tall. What is the percentage increase?
1. Find the Change: How much did it actually grow? \( 15 - 10 = 5\text{cm} \).
2. Identify the Original: What was the starting height? It was 10cm.
3. Put it into the formula: \( \frac{5}{10} \times 100 \).
4. Calculate: \( 0.5 \times 100 = 50\% \).
The plant grew by 50%.
Common Mistake to Avoid: Always divide by the Original (Starting) value, not the new one! Many students accidentally divide by the bigger number or the newest number. Always ask yourself: "What was the value before the change happened?"
Key Takeaway: Difference ÷ Original × 100. Write this formula at the top of your paper during exams!
5. Summary and Quick Tips
The "Cheat Sheet" for Percentage Change:
- To Increase: Multiplier is \( 1 + \text{decimal} \). (e.g., +10% is \(\times 1.1\))
- To Decrease: Multiplier is \( 1 - \text{decimal} \). (e.g., -10% is \(\times 0.9\))
- To find the % change: \( \frac{\text{Difference}}{\text{Original}} \times 100 \)
Don't Forget:
- Units: If the question is about money, make sure your answer has a £ sign. If it asks for a percentage, make sure it has a % sign!
- Sanity Check: If a price goes up, your answer should be higher than the original. If it’s a "sale," it should be lower. It sounds simple, but always check your final answer makes sense in the real world!
- Be Brave: If you get a percentage higher than 100%, that’s okay! It just means the value has more than doubled.
You’ve got this! Percentage change is just a way of telling a story about how numbers move. Keep practicing, and it will become second nature.