Ratios: Relationships in Numbers
A Ratio is a comparison of two numbers by division. It can be written as \(a:b\) or \(\frac{a}{b}\).
Proportions: Equality of Ratios
A Proportion is an equation stating that two ratios are equal. \[\frac{a}{b} = \frac{c}{d}\]
Cross-Multiplication
The most powerful tool for solving proportions is cross-multiplication: \[ad = bc\]
Worked Examples
Example 1: Solving for a Missing Part
Solve for \(x\): \[\frac{x}{10} = \frac{3}{5}\]
- Cross-multiply: \(5x = 30\).
- Divide by 5: \(x = 6\).
- Result: \(x = 6\)
Example 2: Scale Models
If a model car is 1:24 scale and the model is 5 inches long, how long is the real car? \[\frac{1}{24} = \frac{5}{x}\]
- Cross-multiply: \(1x = 24 \times 5 = 120\).
- Result: 120 inches (10 feet).
Example 3: Complex Proportions
Solve: \[\frac{x+2}{4} = \frac{x-1}{2}\]
- Cross-multiply: \(2(x + 2) = 4(x - 1)\).
- Expand: \(2x + 4 = 4x - 4\).
- Subtract \(2x\): \(4 = 2x - 4\).
- Add 4: \(8 = 2x\).
- Divide by 2: \(x = 4\).
- Result: \(x = 4\)
The Bridge to Quantum Mechanics
Physics is the study of ratios. The most famous ratio in quantum mechanics is the Fine Structure Constant (\(\alpha \approx 1/137\)). It is a ratio of physical constants that determines the strength of the electromagnetic interaction. When we calculate the "Probability" of an event, we are essentially calculating a ratio: (desired outcomes) / (total possibilities). If you can't scale a ratio, you can't predict the behavior of a quantum field.