Fractional units

A fraction is a number that represents a part of a whole. The simplest fractions are the fractional units , one half, one third, one fourth, and so on. Master these and the rest of fractions becomes much easier.

Concept

If you take a whole , say a chapati , and cut it into $q$ equal pieces, each piece is called one $q$-th of the chapati. It is written as $\frac{1}{q}$.

• Cut into $2$ equal pieces $\to$ each piece is $\frac{1}{2}$ ("one half").
• Cut into $3$ equal pieces $\to$ each piece is $\frac{1}{3}$ ("one third").
• Cut into $4$ equal pieces $\to$ each piece is $\frac{1}{4}$ ("one fourth" or "one quarter").
• Cut into $10$ equal pieces $\to$ each piece is $\frac{1}{10}$ ("one tenth").

The bottom number is the denominator , it tells how many equal parts make the whole. The top number is the numerator , it tells how many of those parts we are talking about.

If you take $3$ of those $4$ quarter-pieces, you have $\frac{3}{4}$ of the chapati. In general $\frac{p}{q}$ means $p$ of the equal parts when the whole is split into $q$ parts.

One important condition

The pieces must be equal. If you cut a chapati into a tiny sliver and a huge chunk and call each "one half", you have lied. The "$\frac{1}{q}$" only makes sense when all $q$ pieces are the same.

Fractions on a number line

You can also place fractions on a number line. Between $0$ and $1$, the fractional unit $\frac{1}{4}$ marks four equal jumps: $\frac{0}{4}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}$. Note that $\frac{4}{4} = 1$ , when you have all the parts, you have the whole.

Comparing fractional units

Here is the strange but important point: smaller denominator means bigger piece. If you share a chapati between $3$ friends, each gets more than if you share it between $10$ friends. So:

$\frac{1}{2} > \frac{1}{3} > \frac{1}{4} > \frac{1}{5} > \dots > \frac{1}{100}$

The more people, the smaller each share. This is the most common place where beginners trip up , they think a bigger denominator means a bigger fraction. Always picture the chapati.

Worked examples

Example 1. A pizza is cut into $8$ equal slices. Riya eats $3$ slices. What fraction did she eat?

• $3$ slices out of $8 = \frac{3}{8}$.

Example 2. Sketch $\frac{2}{5}$ of a rectangle.

• Draw a rectangle. Divide it into $5$ equal columns. Shade any $2$ of them.

Example 3. Which is larger: $\frac{1}{6}$ or $\frac{1}{8}$?

• Sharing among $6$ gives more than among $8$. So $\frac{1}{6} > \frac{1}{8}$.

Example 4. A clock face is divided by the $12$ numbers into $12$ equal parts. Each part is what fraction of the whole?

• $\frac{1}{12}$.

Example 5. Show $\frac{3}{4}$ on a number line.

• Divide the segment from $0$ to $1$ into $4$ equal pieces.
• Mark the third division. That point is $\frac{3}{4}$.

Example 6. What fraction is shaded if a square is split into $9$ equal smaller squares and $5$ of them are shaded?

• $\frac{5}{9}$.

Try it yourself

1. A cake is cut into $12$ equal pieces. What fraction is one piece? Three pieces?
2. Sketch $\frac{3}{5}$ of a circle (a pie chart).
3. Which is larger: $\frac{1}{4}$ or $\frac{1}{7}$?
4. Place these fractions on a number line: $\frac{1}{5}$, $\frac{3}{5}$, $\frac{4}{5}$.
5. A chocolate bar has $10$ pieces. You eat $7$. What fraction did you eat?
6. Write the fraction "five eighths" in symbols.
7. Can a fraction be zero? When?
8. Investigate: could a fraction be greater than $1$? Sketch what $\frac{5}{4}$ would look like.

Activity

Folding fractions. Take a strip of paper. Fold it in half , each part is $\frac{1}{2}$. Fold again to get $\frac{1}{4}$. Once more for $\frac{1}{8}$. Once more for $\frac{1}{16}$. Mark each fold line and write the fraction on each piece. Show how $\frac{1}{2}$ equals $\frac{2}{4}$ equals $\frac{4}{8}$ equals $\frac{8}{16}$.