Fractions - Types of Fractions (Proper, Improper, Mixed)

A fraction represents a part of a whole, written in the form $\frac{a}{b}$, where $a$ is the numerator (parts being considered) and $b$ is the denominator (total equal parts the whole is divided into). Visually, if a rectangular chocolate bar is divided into $5$ equal pieces and you eat $2$, you have consumed $\frac{2}{5}$ of the bar.

Proper Fractions are fractions where the numerator is smaller than the denominator ($a < b$). The value of a proper fraction is always less than $1$. On a number line, a proper fraction like $\frac{3}{4}$ is located between $0$ and $1$. Visually, it represents less than one full object, like a pizza with a few slices missing.

Improper Fractions have a numerator that is equal to or greater than the denominator ($a \ge b$). Their value is always $1$ or greater than $1$. For example, $\frac{7}{4}$ represents $7$ quarters. Visually, this looks like one whole circle divided into $4$ parts, plus another identical circle with $3$ out of $4$ parts shaded.

Mixed Fractions (or Mixed Numbers) consist of a whole number and a proper fraction written together, such as $2 \frac{1}{3}$. This is an alternative way to represent an improper fraction. Visually, $2 \frac{1}{3}$ represents two completely filled containers and a third container that is only one-third full.

Conversion of Mixed to Improper: To change a mixed fraction to an improper fraction, multiply the whole number by the denominator and add the numerator. This total becomes the new numerator, while the denominator stays the same. For example, $3 \frac{1}{2}$ is visualized as $3$ wholes (which is $6$ halves) plus $1$ half, totaling $\frac{7}{2}$.

Conversion of Improper to Mixed: Divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the divisor remains the denominator. For example, $\frac{11}{4}$ is $11 \div 4$, which gives $2$ wholes and a remainder of $3$, resulting in $2 \frac{3}{4}$.

Fractions on a Number Line: Proper fractions occupy the space between $0$ and $1$. Improper and Mixed fractions occupy the space to the right of $1$. To plot $1 \frac{1}{2}$, you move one full unit from $0$ to $1$, then move half the distance between $1$ and $2$.