Mathematics: Class 6 — Chapter 7: Fractions

When a whole number of things are shared equally among a certain number of people, fractions tell us how much each share is.

• Sharing 1 roti between 2 children: Each gets half, written as $\frac{1}{2}$ (read as "one half" or "one upon two").
• Sharing 1 roti among 4 children: Each gets a quarter, written as $\frac{1}{4}$.
• Since more children share the same roti in the second case, each gets a smaller share: $\frac{1}{2} > \frac{1}{4}$.

When one unit is divided into several equal parts, each part is called a fractional unit or unit fraction. Examples include $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$

We can measure parts of a whole using fractional units. For example, if a whole chikki is divided into 4 equal parts, each part is $\frac{1}{4}$. A piece consisting of 3 such parts is measured as $\frac{3}{4}$ of the chikki.

By folding paper strips, we can visualize and collect fractional units to measure quantities:

• $2 \text{ times } \frac{1}{4} = \frac{2}{4}$
• $3 \text{ times } \frac{1}{4} = \frac{3}{4}$
• $4 \text{ times } \frac{1}{4} = \frac{4}{4} = 1$

In any fraction $\frac{a}{b}$:

• The top number $a$ is the numerator (representing how many fractional units are collected).
• The bottom number $b$ is the denominator (representing the fractional unit $\frac{1}{b}$).

Fractions can be represented as points on a number line by dividing the unit distance between integers into equal parts.

• To mark $\frac{1}{3}$, divide the interval from $0$ to $1$ into 3 equal parts. The first mark represents $\frac{1}{3}$, the second represents $\frac{2}{3}$.
• There are an uncountable number of fractions lying between $0$ and $1$.

Fractions can be classified based on their relationship to 1 unit:

• Fractions less than 1: The numerator is smaller than the denominator (e.g., $\frac{1}{2}, \frac{2}{3}$).
• Fractions greater than 1: The numerator is larger than the denominator (e.g., $\frac{3}{2}, \frac{5}{2}$).

Fractions that represent the same value, length, or share are called equivalent fractions.

• Example: $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$

To compare fractions with different denominators, convert them into equivalent fractions with a common denominator:

1. Find a common multiple of the denominators.
2. Convert each fraction to an equivalent fraction with this common denominator.
3. Compare the numerators.

Example: To compare $\frac{4}{5}$ and $\frac{7}{9}$, use the common denominator $45$:

• $\frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45}$
• $\frac{7}{9} = \frac{7 \times 5}{9 \times 5} = \frac{35}{45}$
• Since $36 > 35$, we have $\frac{4}{5} > \frac{7}{9}$.

Fractions have a rich history originating in ancient India:

• In Sanskrit, a fraction was called bhinna (meaning 'broken'), bhaga, or ansha (meaning 'part' or 'piece').
• The modern notation of writing fractions (numerator over denominator) originated in Indian texts like the Bakshali manuscript (c. 300 CE).
• The formal rules for adding and subtracting fractions were codified by Brahmagupta in his treatise Brahmasphuṭasiddhānta (628 CE).

Egyptian Fractions: Ancient Egyptians expressed general fractions as sums of unique unit fractions.

• Writing 1 as the sum of three different unit fractions: $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$.