Chapter 8 — Working with Fractions

Chapter 8, 'Working with Fractions', covers multiplication of fractions (fraction × whole number, whole number × fraction, and fraction × fraction) and division of fractions using the reciprocal, along with exploring how products and quotients compare in size to the original numbers.

Multiply the whole number by the numerator of the fraction and keep the denominator unchanged. For example, 3 × (1/4) = 3/4. Equivalently, you can first divide the multiplicand by the denominator of the multiplier, then multiply the result by the numerator.

Brahmagupta's formula, first stated in his Brahmasphutasiddhanta in 628 CE, says: (a/b) × (c/d) = (a×c)/(b×d). That is, multiply the numerators together and the denominators together.

If the sides of a rectangle are fractions, the area equals the product of those fractional sides. Using a unit square as the whole, dividing it into rows and columns equal to the denominators of the two fractions visually shows why (a/b) × (c/d) = (a×c)/(b×d).

Before multiplying, you can divide any numerator and any denominator by their common factor, reducing the result to its lowest form without extra simplification steps. For example, (12/7) × (5/24): since 12 and 24 share a factor of 12, dividing both by 12 gives (1/7) × (5/2) = 5/14. This process of reducing to lowest terms is called apavartana in Indian mathematics.

The reciprocal of a fraction (a/b) is (b/a). Multiplying a fraction by its reciprocal gives 1. To divide two fractions, multiply the dividend by the reciprocal of the divisor: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). This rule was stated by Brahmagupta in 628 CE and clarified by Bhaskara II in his Lilavati (1150 CE).

Not always. When both numbers being multiplied are between 0 and 1, the product is less than both. When one number is between 0 and 1 and the other is greater than 1, the product is less than the larger number but greater than the smaller one. When both numbers are greater than 1, the product is greater than both.

When the divisor is between 0 and 1, the quotient is greater than the dividend. For example, 6 ÷ (1/4) = 24, which is greater than 6. When the divisor is greater than 1, the quotient is less than the dividend.

No. The order of multiplication does not matter: (a/b) × (c/d) = (c/d) × (a/b). This is also consistent with Brahmagupta's formula, since a×c = c×a and b×d = d×b.

The chapter traces fraction arithmetic to ancient India. The Shulbasutras (c. 800 BCE) used non-unit fractions for geometric constructions. By 150 BCE, reduction of fractions to lowest terms (apavartana) was mentioned in philosophical writings of Umasvati. General rules for all four operations on fractions were codified by Brahmagupta in his Brahmasphutasiddhanta (628 CE). Bhaskara II further clarified division of fractions in his Lilavati (1150 CE). The Indian theory was later transmitted to Arab, African, and eventually European mathematicians.

Brahmagupta (628 CE) first stated the general formula for multiplying fractions — product of numerators over product of denominators — and the rule for dividing fractions by interchanging the numerator and denominator of the divisor, in his work Brahmasphutasiddhanta.

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