Ratios, Proportions, and Real-Life Applications
Introduction
Ratios and proportions are vital for solving problems in GCSE Maths and beyond. They’re practical, appearing in real-life contexts like recipes, map reading, and unit conversions.
This article will cover:
- The basics of ratios and proportions.
- Practical real-life applications.
- Strategies to tackle ratio problems effectively in exams.
Understanding Ratios
Simplifying Ratios
Ratios can be simplified like fractions by dividing all terms by their highest common factor (HCF).
Example: Simplify \( 12 : 18 : 24 \).
- HCF = 6.
- Simplify: \( \frac{12}{6} : \frac{18}{6} : \frac{24}{6} = 2 : 3 : 4 \).
Sharing Amounts in a Given Ratio
Split amounts proportionally based on the ratio.
Example: Share £120 in the ratio \( 2 : 3 \).
- Total parts = \( 2 + 3 = 5 \).
- One part = \( \frac{120}{5} = 24 \).
- Shares: \( 2 \times 24 = 48 \), \( 3 \times 24 = 72 \).
Understanding Proportions
Proportions compare two ratios and can be direct or inverse.
Direct Proportion
Quantities increase or decrease together.
Example: If 5 apples cost £10, how much will 8 apples cost?
- Set up proportion: \( \frac{5}{10} = \frac{8}{x} \).
- Solve: \( 5x = 80 \implies x = 16 \).
Inverse Proportion
One quantity increases while the other decreases.
Example: If 3 people take 12 hours to paint a house, how long will 6 people take?
- Use inverse proportion formula: \( 3 \times 12 = 6 \times x \).
- Solve: \( 36 = 6x \implies x = 6 \).
Real-Life Applications
Recipes:
- Scale ingredients up or down based on serving size.
Example: A recipe for 4 people uses 200g of flour. How much is needed for 6 people?
\[
\text{Flour} = \frac{200}{4} \times 6 = 300g.
\]Maps and Scales:
- Use ratios to interpret map distances.
Example: On a map with a scale of \( 1 : 50000 \), 2cm represents \( 2 \times 50000 = 100000 \)cm, or 1km.
Currency Conversion:
- Convert amounts using exchange rates.
Example: £1 = €1.20. How many euros is £50?
\[
50 \times 1.20 = 60 \, \text{euros}.
\]
Practice Question
Question: Divide £180 in the ratio \( 3 : 5 \).
Solution:
- Total parts = \( 3 + 5 = 8 \).
- One part = \( \frac{180}{8} = 22.50 \).
- Shares: \( 3 \times 22.50 = 67.50 \), \( 5 \times 22.50 = 112.50 \).
Conclusion
Mastering ratios and proportions is essential for solving real-life problems in GCSE Maths. Practise these concepts regularly to gain confidence in your exams.
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