A fraction expresses a number as a quotient. When a rectangle is split into four equal areas and three are shaded, the denominator 4 counts the equal parts in one whole and the numerator 3 counts the selected parts. The same number marks a position three quarters of the way from zero to one on a number line. The denominator cannot be zero.
When the numerator reaches or exceeds the denominator
For positive fractions, a numerator smaller than the denominator gives a proper fraction. An improper fraction has numerator at least as large as its denominator. Some improper fractions represent whole numbers because the numerator is a multiple of the denominator. A mixed number writes the whole units and the remaining fraction together. A unit fraction has numerator 1. A decimal fraction has a denominator such as 10 or 100. A complex fraction has another fraction in its numerator or denominator.
Convert eleven quarters into a mixed number
Divide 11 by 4. The quotient 2 counts the complete wholes. The remainder 3 counts the quarters still left. To convert back, multiply the whole number by the denominator and add the fractional numerator. Here two wholes contribute eight quarters, and three more make eleven.
Reduce first, then compare in the same units
To simplify 18 over 24, divide both numbers by their greatest common divisor, 6. Three and four have no common factor greater than 1, so the resulting fraction is in simplest form. To compare three quarters with five sixths, write both in twelfths. Nine twelfths is smaller than ten twelfths. Comparing just the numerators makes sense only when both denominators are equal and positive.
Two drawings can show the same fraction while using different sizes of whole. To compare the actual amount of material, check that the wholes have the same size. For the numerical relationship between fractions, see equivalent fractions.
