Fraction
$$\frac{\text{Numerator}}{\text{Denominator}}$$
What is Fraction?
A fraction represents a part of a whole as a ratio of two integers: a numerator (top) and a denominator (bottom), written $$\frac{a}{b}$$ where $$b \neq 0$$. The denominator defines the size of each equal part; the numerator counts how many of those parts are taken. Fractions are exact — unlike decimals, which may require infinite digits to express a value like $$\frac{1}{3}$$, a fraction represents the quantity precisely without rounding.
Fractions are classified by the relationship between numerator and denominator. A proper fraction has a numerator smaller than the denominator ($$\frac{3}{4}$$) and represents a value less than 1. An improper fraction has a numerator equal to or greater than the denominator ($$\frac{7}{4}$$) and represents 1 or more. A mixed number combines a whole-number part with a proper fraction ($$1\frac{3}{4}$$) and is the human-readable form of an improper fraction.
Arithmetic with fractions requires attention to the denominator. Multiplication and division operate directly on numerators and denominators. Addition and subtraction require a common denominator first — the least common denominator (LCD) is the smallest integer divisible by all denominators in the expression.
When to use Fraction
Use fractions when exact rational representation is required and rounding is unacceptable — engineering tolerances, probability calculations, and algebraic manipulation all benefit from fractions over decimals. Use decimals when inputting values into digital systems or when communicating measurements where decimal notation is the convention.
Worked examples
| Type | Example | Decimal | Notes |
|---|---|---|---|
| Proper fraction | 3/4 | 0.75 | Numerator < denominator; value < 1 |
| Improper fraction | 9/4 | 2.25 | Numerator ≥ denominator; value ≥ 1 |
| Mixed number | 2 1/4 | 2.25 | Equivalent to 9/4 |
| Unit fraction | 1/7 | 0.142857… | Numerator = 1; repeating decimal |
| Equivalent fractions | 2/3 = 4/6 = 8/12 | 0.6666… | Same value, different form |
Common pitfalls
Fractions with different denominators cannot be added or subtracted directly. Adding $$\frac{1}{3} + \frac{1}{4}$$ is not $$\frac{2}{7}$$ — the correct answer is $$\frac{7}{12}$$. The LCD of 3 and 4 is 12, giving $$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$. This error is especially common when students apply the addition-of-numerators intuition to denominators as well.
Frequently asked questions
How do I add fractions with different denominators?
Find the least common denominator (LCD) of all fractions. Convert each fraction to an equivalent with the LCD as denominator by multiplying numerator and denominator by the appropriate factor. Then add the numerators and keep the LCD. For 1/4 + 1/6: LCD = 12, so 3/12 + 2/12 = 5/12.
How do I divide two fractions?
Multiply the first fraction by the reciprocal of the second: a/b ÷ c/d = a/b × d/c = (a×d)/(b×c). For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
What is the difference between a fraction and a ratio?
A fraction expresses a part of a whole — the denominator is the total number of equal parts. A ratio compares two separate quantities — the denominator is not the total. A fraction 3/5 says "3 out of 5 equal parts of the whole"; a ratio 3:2 says "3 of one type for every 2 of another."