Mathematics: Class 7 (Part 1) — Chapter 8: Working with Fractions

Multiplication can be understood as repeated addition. When multiplying a whole number by a fraction, we extend this concept:

• For example, if a tortoise walks $\frac{1}{4}\text{ km}$ in $1\text{ hour}$, the distance covered in $3\text{ hours}$ is calculated as: $$3 \times \frac{1}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}\text{ km}$$
• General Rule: To multiply a fraction by a whole number, we multiply the numerator by the whole number and keep the denominator the same: $$\frac{a}{b} \times c = \frac{a \times c}{b}$$

When multiplying two fractions, we can visualize the operation using an area model of a unit square:

• The area of a rectangle with fractional sides represents the product of those fractions.
• General Formula: To multiply two fractions, multiply their numerators together and their denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
• This general formula was historically codified by the Indian mathematician Brahmagupta in his work Brāhmasphuṭasiddhānta (628 CE).
• Simplification (Apavartana): Before multiplying, it is often easier to divide the numerators and denominators by their common factors, a process known historically as apavartana.

• Order of Multiplication: The order in which two fractions are multiplied does not affect the product (Commutative Property): $$\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}$$
• Product Size Relationships:
  • If both fractions are greater than $1$, the product is greater than both.
  • If both fractions are between $0$ and $1$, the product is less than both.
  • If one fraction is between $0$ and $1$ and the other is greater than $1$, the product lies between the two numbers.

To understand division, we first define the reciprocal:

• Reciprocal: Two non-zero numbers are reciprocals of each other if their product is $1$.
• The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$ because: $$\frac{a}{b} \times \frac{b}{a} = 1$$

Division can be restated as finding a missing multiplier. This leads to the rule for dividing fractions:

• Division Rule: To divide one fraction by another, multiply the dividend by the reciprocal of the divisor: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$
• This method was also explicitly formulated by Brahmagupta and later clarified by Bhāskara II in his mathematical treatise Līlāvatī (1150 CE).
• Quotient Size Relationships:
  • If the divisor is between $0$ and $1$, the quotient is greater than the dividend (e.g., $6 \div \frac{1}{4} = 24$, where $24 > 6$).
  • If the divisor is greater than $1$, the quotient is less than the dividend.

Fractions are widely used to solve real-world sharing and rate problems:

• Rate Problems: Calculating individual shares when quantities are distributed (e.g., finding the volume of milk per cup of tea when a total volume is divided among multiple cups).
• Work and Time Problems: Determining combined rates of work, such as multiple fountains filling a cistern at different fractional daily rates.
• Historical Geometry: The ancient Indian text Śhulbasūtra (c. 800 BCE) contains early examples of dividing areas using fractional brick dimensions.

• Geometric Fractional Relations: Determining the shaded fraction of a complex geometric shape by breaking it down into nested sub-regions (e.g., finding the fraction of a unit square occupied by a shaded triangle).
• Historical Currency Problems: A classic problem from Bhāskara II's Līlāvatī involves calculating a tiny fractional donation of a historical silver coin (dramma) down to the level of individual cowrie shells, demonstrating the practical application of nested fraction multiplication.