Fractions/decimals/percentages; ratio/proportion - Mathematics IGCSE Study Notes
Overview
Imagine you're sharing a pizza with friends, or trying to figure out a discount in a shop, or even mixing juice for a party. All these everyday situations use **Fractions, Decimals, Percentages, Ratios, and Proportions**! These are all just different ways of talking about parts of a whole, or comparing amounts. Learning these topics isn't just for passing your IGCSE exam; it's about understanding the world around you. From cooking to shopping to understanding news reports, these mathematical tools help you make sense of numbers in real life. They help you compare things fairly and understand how different amounts relate to each other. So, let's dive in and make these seemingly tricky ideas super simple. We'll break down how they work, how they connect, and how you can use them like a pro!
What Is This? (The Simple Version)
Think of it like this: you have a whole cake. You can talk about a piece of that cake in different ways:
- Fraction: This is like saying you ate "1 out of 4 slices" or 1/4 of the cake. It's a way to show a part divided by a whole, using a line in between two numbers.
- Decimal: This is another way to show a part of the whole, but using a decimal point. So, 1/4 of the cake is the same as 0.25 of the cake. It's like saying twenty-five hundredths.
- Percentage: This means "out of 100". So, 1/4 of the cake is the same as 25% of the cake. Imagine the whole cake is 100 small pieces, and you ate 25 of them.
They are all just different languages for the same idea: a part of a whole!
Now, let's add Ratio and Proportion.
- Ratio: This is like comparing two different things. If you mix juice, and you use 1 cup of concentrate for every 3 cups of water, the ratio of concentrate to water is 1:3. It's a way to show how two amounts relate to each other.
- Proportion: This means that two ratios are equal. If you double your juice recipe, you'd use 2 cups of concentrate to 6 cups of water. The ratio 2:6 is proportional to 1:3 because they both simplify to the same comparison.
Real-World Example
Imagine you're at a shop and you see a T-shirt that originally costs $20, but it has a 25% discount.
Let's break this down:
- Understand the Percentage: 25% means 25 out of every 100. So, 25% of the price is what you save.
- Convert to Decimal (Optional but helpful): To calculate with percentages, it's often easiest to change them into a decimal. Remember, percentage means "out of 100", so 25% is 25 divided by 100, which is 0.25.
- Calculate the Discount Amount: Multiply the original price by the decimal: $20 * 0.25 = $5. So, you save $5.
- Find the New Price: Subtract the discount from the original price: $20 - $5 = $15. The T-shirt now costs $15.
See how percentages help you quickly figure out how much you save and what the new price is? This is super useful for shopping!
How It Works (Step by Step)
Let's learn how to switch between these different forms, like changing languages for the same idea. **Converting between Fractions, Decimals, and Percentages:** 1. **Fraction to Decimal:** Divide the top number (numerator) by the bottom number (denominator). For example, 3/4 becomes 3 ÷ 4 = 0.75....
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Key Concepts
- Fraction: A way to represent a part of a whole, written as one number over another, like 1/2.
- Decimal: A way to represent a part of a whole using a decimal point, like 0.5.
- Percentage: A way to represent a part of a whole as a number out of 100, followed by a % sign, like 50%.
- Numerator: The top number in a fraction, showing how many parts you have.
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Exam Tips
- →Always read the question carefully to see if the answer needs to be a fraction, decimal, or percentage, and convert accordingly.
- →When working with percentages, remember that 'of' usually means 'multiply'. For example, '20% of 50' means 0.20 * 50.
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