Number - Proper and Improper Fractions on the Number Line

Represent the proper fraction $\frac{3}{4}$ on a number line.

1. Draw a number line and mark $0$ and $1$.
2. Divide the segment between $0$ and $1$ into $4$ equal parts (since the denominator is $4$).
3. Starting from $0$, count $3$ marks to the right.
4. The third mark represents $\frac{3}{4}$.

Since $\frac{3}{4}$ is a proper fraction, it must lie between $0$ and $1$. Dividing the unit into $4$ parts gives segments of $\frac{1}{4}, \frac{2}{4}, \frac{3}{4}$, and $\frac{4}{4}$ (which is $1$).

Locate the improper fraction $\frac{11}{4}$ on the number line.

1. Convert $\frac{11}{4}$ to a mixed number: $11 \div 4 = 2$ with a remainder of $3$. So, $\frac{11}{4} = 2 \frac{3}{4}$.
2. This point lies between the whole numbers $2$ and $3$.
3. Divide the space between $2$ and $3$ into $4$ equal parts.
4. The point $\frac{11}{4}$ is the $3$rd mark after the number $2$.

Converting to a mixed number helps identify which two integers the improper fraction falls between. $\frac{11}{4}$ is greater than $2$ but less than $3$.

Identify the fraction represented by point $P$ if it is at the $2$nd mark between $1$ and $2$, where the interval is divided into $5$ equal parts.

1. The interval is between $1$ and $2$, so the whole number part is $1$.
2. The denominator is $5$ because there are $5$ equal parts.
3. The numerator is $2$ because it is at the $2$nd mark.
4. Point $P = 1 \frac{2}{5}$.
5. As an improper fraction: $1 \frac{2}{5} = \frac{(1 \times 5) + 2}{5} = \frac{7}{5}$.

The position $1 \frac{2}{5}$ indicates we have passed the whole number $1$ and moved an additional $\frac{2}{5}$ towards $2$.