Simplification of fractions
To simplify a fraction means to make it simpler by reducing numerator and denominator by the same factor. The fraction thus takes a shorter form but still represents the same number.
Example 1: A simple fraction
Simplify the fraction:
$$ \large \frac{6}{8} $$
Both 6 and 8 can be divided by 2:
$$ \large \frac{6 : 2}{8 : 2} = \frac{3}{4} $$
The fraction is simplified to \( \frac{3}{4} \).
Example 2: Larger numbers
Simplify the fraction:
$$ \large \frac{42}{56} $$
Here we can find the greatest common divisor (GCD) of 42 and 56, which is 14:
$$ \large \frac{42 : 14}{56 : 14} = \frac{3}{4} $$
The fraction is simplified to \( \frac{3}{4} \).
Example 3: Fractions with letters
Simplify the fraction:
$$ \large \frac{12a}{18a} $$
Both numerator and denominator have a common factor 6 and a common factor \(a\):
$$ \large \frac{12a : 6a}{18a : 6a} = \frac{2}{3} $$
The fraction is simplified to \( \frac{2}{3} \).
Example 4: Negative sign
Simplify the fraction:
$$ \large \frac{-15}{20} $$
Both 15 and 20 can be divided by 5:
$$ \large \frac{-15 : 5}{20 : 5} = \frac{-3}{4} $$
The fraction is simplified to \( -\frac{3}{4} \).
Example 5: Fractions with several factors
Simplify the fraction:
$$ \large \frac{18xy}{24x} $$
We can reduce both by 6 and by \(x\):
$$ \large \frac{18xy : 6x}{24x : 6x} = \frac{3y}{4} $$
The fraction is simplified to \( \frac{3y}{4} \).
Summary
When you simplify fractions, remember:
- Find a common factor for numerator and denominator.
- Divide both numerator and denominator by the same factor.
- The fraction still represents the same number but is written in simpler form.
- You can simplify fractions with numbers, letters, or both.
Simplification of fractions is a fundamental skill in mathematics, used in arithmetic, algebra and later in equations and functions.