Dividing is Just Multiplication in Disguise
To divide by a fraction, you use the "Keep, Change, Flip" method:
- Keep the first fraction.
- Change the division sign to multiplication.
- Flip the second fraction (this is called the Reciprocal).
\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]
Worked Examples
Example 1: Basic Division
Evaluate: \(\frac{1}{2} \div \frac{3}{4}\)
- Keep \(\frac{1}{2}\).
- Change \(\div\) to \(\times\).
- Flip \(\frac{3}{4}\) to \(\frac{4}{3}\).
- Multiply: \(\frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}\).
- Result: \(\frac{2}{3}\)
Example 2: Dividing by a Whole Number
Evaluate: \(\frac{5}{6} \div 3\)
- Think of 3 as \(\frac{3}{1}\).
- Flip to \(\frac{1}{3}\).
- Multiply: \(\frac{5}{6} \times \frac{1}{3} = \frac{5}{18}\).
- Result: \(\frac{5}{18}\)
Example 3: Complex Fraction (The Double-Decker)
Evaluate: \[\frac{\frac{2}{5}}{\frac{1}{10}}\]
- This just means \(\frac{2}{5} \div \frac{1}{10}\).
- Flip and multiply: \(\frac{2}{5} \times \frac{10}{1}{}\).
- Cross-cancel 10 and 5 \(\to\) 2 and 1.
- Multiply: \(\frac{2}{1} \times \frac{2}{1} = 4\).
- Result: 4
The Bridge to Quantum Mechanics
In the Schrödinger Equation, we often see the term \(\frac{\hbar^2}{2m}\). This is a "double-decker" fraction because \(\hbar\) itself is \(h / 2\pi\). To solve for energy, you must be comfortable flipping and multiplying these constants. If you get confused by reciprocal multiplication, your units will be wrong, and your electron will "fly off" to the wrong side of the atom!