Adding Parts of Different Sizes
You cannot add \(\frac{1}{2}\) and \(\frac{1}{3}\) directly because the "pieces" are different sizes. It's like trying to add 1 apple and 1 orange—you just have "2 pieces of fruit," but not a single type. To add them, we must make the "pieces" (the denominators) the same.
The Least Common Denominator (LCD)
The LCD is the smallest number that both denominators can divide into. For 2 and 3, the LCD is 6.
The Addition Process
- Find the LCD.
- Convert both fractions to have that LCD using the multiplication rule (Lesson 4).
- Add the numerators. Keep the denominator the same.
Worked Examples
Example 1: Basic Addition
Evaluate: \(\frac{1}{4} + \frac{2}{4}\)
- The denominators are already the same.
- Add the tops: \(1 + 2 = 3\).
- Result: \(\frac{3}{4}\)
Example 2: Different Denominators
Evaluate: \(\frac{1}{2} + \frac{1}{3}\)
- LCD is 6.
- Convert \(\frac{1}{2}\): Multiply top and bottom by 3 \(\to \frac{3}{6}\).
- Convert \(\frac{1}{3}\): Multiply top and bottom by 2 \(\to \frac{2}{6}\).
- Add: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).
- Result: \(\frac{5}{6}\)
Example 3: Mixed Signs
Evaluate: \(\frac{3}{4} - \frac{5}{6}\)
- LCD of 4 and 6 is 12.
- \(\frac{3}{4} = \frac{9}{12}\).
- \(\frac{5}{6} = \frac{10}{12}\).
- Subtract: \(\frac{9}{12} - \frac{10}{12} = \frac{-1}{12}\).
- Result: \(-\frac{1}{12}\)
The Bridge to Quantum Mechanics
Quantum Mechanics is built on the Principle of Superposition. When two quantum states overlap, we "add" them together. Often, these states are fractions of a whole. If state A has a \(\frac{1}{2}\) chance of happening and state B has a \(\frac{1}{3}\) chance, the math of finding the total state is identical to what you just did. If you can't find a common denominator, you can't predict where a particle will be.