A Fraction of a Fraction Is Smaller Still Working with Fractions

Aaron walks at 3 km per hour; his pet tortoise walks at only 1/4 km per hour. In 3 hours, the tortoise covers 3 x 1/4 = 3/4 km -- a whole number times a fraction, found simply by repeated addition (1/4+1/4+1/4). But what does it mean to walk for only 1/2 an hour at the tortoise's already-fractional pace? This needs a genuinely new operation: multiplying one fraction by another, 1/2 x 1/4, and a unit square split into a 2x4 grid makes the answer visible directly -- shading half the square one way and a quarter the other way, the doubly-shaded overlap covers exactly 1 out of 8 equal pieces, giving 1/2 x 1/4 = 1/8.

Generalising the unit-square picture for any two fractions gives a clean rule: multiply the two numerators together, and multiply the two denominators together, (a/b) x (c/d) = (ac)/(bd). Before multiplying out large numerators and denominators, cancelling any common factor shared between a numerator and a denominator first, a technique with the Sanskrit name apavartana, keeps the numbers small and the arithmetic simple -- 12/7 x 5/24 simplifies before multiplying at all, since 12 and 24 share a factor, turning an otherwise-large calculation into an easy one.

Dividing by a fraction, like 1 divided by 2/3, is really asking a multiplication question backwards: what times 2/3 gives 1? Since 2/3 x 3/2 = 1, the answer is 3/2. This 'flip and multiply' shortcut always works: dividing by any fraction is exactly the same as multiplying by that fraction's reciprocal (its numerator and denominator swapped), since a fraction multiplied by its own reciprocal always equals exactly 1. Brahmagupta's Brahmasphutasiddhanta (628 CE) is the first known source to state both the multiplication rule and this reciprocal-based division rule explicitly, in the exact form still used today.

Leena makes 5 cups of tea using 1/4 litre of milk total; dividing 1/4 by 5 (i.e. multiplying by the reciprocal 1/5) gives 1/20 litre of milk in each cup. Baudhayana's Shulbasutra (around 800 BCE) poses a genuinely ancient version of the same idea: covering an area of 7 and 1/2 square units using square bricks of side 1/5 unit needs dividing the total area by one brick's own area (1/5 x 1/5 = 1/25), giving 375/2 bricks. Chaturveda Prithudakasvami (around 860 CE), commenting on Brahmagupta's own book, poses a rate problem: four fountains fill a cistern alone in 1, 1/2, 1/4, and 1/5 of a day respectively; combining their rates (found by dividing 1 by each time) gives 1+2+4+5=12 cisterns' worth filled per day together, so the full cistern takes exactly 1/12 of a day.

Bhaskaracharya's Lilavati (1150 CE) tells of a miser who, pressed to give a beggar something, hands over a cascading, deliberately shrinking fraction of a single dramma coin: half, of two-thirds, of three-quarters, of one-fifth, of one-sixteenth, of one-quarter. Multiplying this entire chain of fractions together gives exactly 6/7680, which simplifies to 1/1280 -- and since one dramma coin was itself worth exactly 1280 cowrie shells (the smallest coin denomination of the era), the miser's elaborately fraction-heavy 'generosity' amounts to precisely one single cowrie shell, the smallest possible coin in the entire currency system. Bhaskaracharya's own evident humour is doing real mathematical work here: a long chain of fraction multiplication shrinks a quantity dramatically, however innocent each individual fraction looks on its own.

This chapter's fraction rules carry a genuine, traceable lineage: the Shulbasutras (around 800 BCE) pose early fraction problems; Umasvati (around 150 CE), a Jain scholar, mentions the idea of reducing to lowest terms even in a non-mathematical work; Bhaskara I's own commentary (629 CE) gives a geometric interpretation of fraction multiplication; Sridharacharya (around 750 CE) and Mahaviracharya (around 850 CE) both refined fraction methods further; and the notation eventually reached the Moroccan mathematician Al-Hassar around 1192 CE, before making its way into wider European use only from around the 17th century onward.