Multiplying is Easier than Adding
Unlike addition, multiplication does not require a common denominator. You simply multiply across.
\[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]
Cross-Cancellation: The Secret to Speed
Before you multiply, look for numbers on the top of one fraction and the bottom of another that share a factor. You can simplify them immediately to keep your numbers small.
Worked Examples
Example 1: Basic Multiplication
Evaluate: \(\frac{2}{3} \times \frac{4}{5}\)
- Multiply tops: \(2 \times 4 = 8\).
- Multiply bottoms: \(3 \times 5 = 15\).
- Result: \(\frac{8}{15}\)
Example 2: Cross-Cancellation
Evaluate: \(\frac{5}{8} \times \frac{4}{15}\)
- Top 5 and bottom 15: Divide both by 5 \(\to\) becomes 1 and 3.
- Top 4 and bottom 8: Divide both by 4 \(\to\) becomes 1 and 2.
- Now multiply the small numbers: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).
- Result: \(\frac{1}{6}\)
Example 3: Multiplying a Whole Number
Evaluate: \(6 \times \frac{3}{4}\)
- Think of 6 as \(\frac{6}{1}\).
- Cross-cancel 6 and 4 (divide by 2): becomes 3 and 2.
- Multiply: \(\frac{3}{1} \times \frac{3}{2} = \frac{9}{2}\).
- Result: \(\frac{9}{2}\) or \(4.5\)
The Bridge to Quantum Mechanics
In physics, we often multiply a Physical Constant (like Planck's constant \(h\)) by a measured value. These constants are usually written as fractions or decimals. When we calculate the energy of a photon \(E = hf\), we are essentially multiplying fractions together. Knowing how to cancel terms across equations is the difference between an elegant solution and a mathematical mess.