Understanding Fractions
Fractions represent parts of a whole and consist of a numerator (the number above the line) and a denominator (the number below the line). The numerator represents how many parts we have, while the denominator represents the total number of equal parts that make up a whole.
Basic Fraction Operations
Adding Fractions
To add fractions with the same denominator, add the numerators and keep the denominator the same:
To add fractions with different denominators, first find a common denominator, then add the numerators:
Example: \(\frac{1}{4} + \frac{2}{3}\)
First, find the common denominator: 12
\(\frac{1}{4} = \frac{3}{12}\) and \(\frac{2}{3} = \frac{8}{12}\)
\(\frac{3}{12} + \frac{8}{12} = \frac{11}{12}\)
Subtracting Fractions
Similar to addition, with same denominators, subtract the numerators:
With different denominators, find a common denominator first, then subtract.
Multiplying Fractions
To multiply fractions, multiply the numerators together and multiply the denominators together:
Example: \(\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}\)
Dividing Fractions
To divide fractions, multiply the first fraction by the reciprocal of the second fraction:
Example: \(\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} = 1\frac{7}{8}\)
Simplifying Fractions
A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction:
- Find the greatest common divisor (GCD) of the numerator and denominator
- Divide both the numerator and denominator by the GCD
Example: Simplify \(\frac{8}{12}\)
GCD of 8 and 12 is 4
\(\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\)
Mixed Numbers and Improper Fractions
An improper fraction has a numerator greater than or equal to its denominator. A mixed number combines a whole number with a proper fraction.
Converting Improper Fractions to Mixed Numbers
Divide the numerator by the denominator. The quotient is the whole number part, and the remainder is the numerator of the fractional part.
Example: Convert \(\frac{17}{5}\) to a mixed number
17 ÷ 5 = 3 remainder 2
\(\frac{17}{5} = 3\frac{2}{5}\)
Converting Mixed Numbers to Improper Fractions
Multiply the whole number by the denominator, add the numerator, and put this sum over the original denominator.
Example: Convert \(2\frac{3}{4}\) to an improper fraction
(2 × 4) + 3 = 11
\(2\frac{3}{4} = \frac{11}{4}\)
Tips for Working with Fractions
- Always check if your answer can be simplified further
- When adding or subtracting fractions, always find a common denominator first
- Remember that dividing by a fraction is the same as multiplying by its reciprocal
- To find equivalent fractions, multiply or divide both the numerator and denominator by the same number
- When dealing with mixed numbers, it's often easier to convert them to improper fractions first