What is a Fraction?
A fraction represents a part of a whole. \[\frac{numerator}{denominator} = \frac{parts\_we\_have}{total\_parts\_in\_whole}\]
Equivalent Fractions
The most important rule of fractions: If you multiply or divide the top and bottom by the same number, the value doesn't change. \[\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}\]
Simplifying (Reducing)
To simplify, find the Largest Common Factor of the top and bottom and divide both by it.
Worked Examples
Example 1: Finding an Equivalent Fraction
Convert \(\frac{3}{5}\) into a fraction with a denominator of 20.
- What do we multiply 5 by to get 20? \(5 \times 4 = 20\).
- Do the same to the top: \(3 \times 4 = 12\).
- Result: \(\frac{12}{20}\)
Example 2: Simplifying a Large Fraction
Simplify \(\frac{24}{36}\).
- Find a number that goes into both. They are both divisible by 6. \(\frac{24\div 6}{36\div 6} = \frac{4}{6}\).
- Can we go further? Yes, both are divisible by 2. \(\frac{4\div 2}{6\div 2} = \frac{2}{3}\).
- Result: \(\frac{2}{3}\)
Example 3: Mixed Numbers to Improper Fractions
Convert \(2 \frac{1}{3}\) to a pure fraction.
- 2 wholes is the same as 6 thirds (\(\frac{6}{3}\)).
- Add the extra third: \(\frac{6}{3} + \frac{1}{3} = \frac{7}{3}\).
- Shortcut: (Whole \(\times\) Bottom) + Top = \(2 \times 3 + 1 = 7\). Put it over the bottom.
- Result: \(\frac{7}{3}\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, we don't say a particle is "at position X." we say there is a Probability. A probability is essentially a fraction. If the probability is \(\frac{1}{4}\), it means that if we ran the experiment 4 times, we would find the particle there 1 time on average. Understanding how to scale and simplify these fractions is how we calculate the strength of electron bonds in molecules.