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Fraction

Fraction

A ratio of numbers or variables. Fractions may not have denominator 0.

 

Examples           The fraction 5/8, with 5 as the numerator and 8 as the denominator.

The fraction √5 divided by 2

The fraction (x minus 3) divided by (2x plus 1), with numerator x−3 and denominator 2x+1.

 

 

 

See also

Compound fraction, improper fraction, rational expression, numerator

Key Formula

ab\frac{a}{b}
Where:
  • aa = The numerator — the number on top, representing how many parts you have
  • bb = The denominator — the number on the bottom, representing how many equal parts the whole is divided into (cannot be 0)

Worked Example

Problem: Add the fractions 2/3 and 1/4.
Step 1: Find the least common denominator (LCD) of 3 and 4. The LCD is 12.
LCD(3,4)=12\text{LCD}(3, 4) = 12
Step 2: Rewrite each fraction with denominator 12. Multiply the numerator and denominator of 2/3 by 4, and those of 1/4 by 3.
23=2×43×4=812,14=1×34×3=312\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}, \quad \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
Step 3: Add the numerators and keep the common denominator.
812+312=1112\frac{8}{12} + \frac{3}{12} = \frac{11}{12}
Step 4: Check whether the result can be simplified. Since 11 is prime and does not divide 12, the fraction is already in simplest form.
1112\frac{11}{12}
Answer: 2/3 + 1/4 = 11/12

Another Example

Problem: Simplify the fraction 18/24 to its lowest terms.
Step 1: Find the greatest common factor (GCF) of 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The GCF is 6.
GCF(18,24)=6\text{GCF}(18, 24) = 6
Step 2: Divide both the numerator and the denominator by the GCF.
1824=18÷624÷6=34\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}
Answer: 18/24 simplified is 3/4.

Frequently Asked Questions

What is the difference between a proper fraction and an improper fraction?
A proper fraction has a numerator smaller than its denominator (like 3/5), so its value is less than 1. An improper fraction has a numerator greater than or equal to its denominator (like 7/4), so its value is 1 or greater. Improper fractions can be converted to mixed numbers — for example, 7/4 = 1 3/4.
Why can't the denominator of a fraction be zero?
Division by zero is undefined in mathematics. If the denominator were 0, the fraction would ask "how many groups of size 0 fit into the numerator?" — a question with no meaningful answer. That is why every fraction requires a nonzero denominator.

Fraction vs. Decimal

A fraction expresses a value as a ratio of two integers (e.g., 3/4), while a decimal uses place value and a decimal point (e.g., 0.75). Some fractions produce terminating decimals; others, like 1/3, produce repeating decimals (0.333…).

Why It Matters

Fractions are foundational for nearly every branch of mathematics, from algebra and probability to calculus. In everyday life you use fractions when measuring ingredients, splitting bills, or interpreting statistics like "3 out of 5 voters." Mastering fractions also prepares you for working with ratios, proportions, and rational expressions in more advanced courses.

Common Mistakes

Mistake: Adding fractions by adding the numerators and denominators separately (e.g., writing 1/2 + 1/3 = 2/5).
Correction: You must find a common denominator first. The correct calculation is 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Mistake: Forgetting to simplify fractions to lowest terms after performing an operation.
Correction: Always check whether the numerator and denominator share a common factor greater than 1. For example, after getting 4/8, divide both by 4 to obtain 1/2.

Related Terms