Working with Fractions

Multiplying a whole number by a fraction is really no different from any other multiplication: it is still repeated addition, just with a fractional amount being repeated. Suppose a tap fills 3/4 of a bucket every minute. How much does it fill in 5 minutes? Since each minute adds another 3/4 of a bucket, the total filled is 3/4 + 3/4 + 3/4 + 3/4 + 3/4, which is exactly the same as 5 × 3/4.

To actually compute 5 × 3/4, multiply the whole number directly into the numerator and keep the denominator unchanged: 5 × 3/4 = (5×3)/4 = 15/4, which as a mixed number is 3 3/4. So the tap fills 3 3/4 buckets in 5 minutes.

The same idea works the other way around too — multiplying a fraction by a whole number, like 3/4 × 5, gives exactly the same answer, 15/4, because multiplying fractions (like multiplying whole numbers) doesn't care which order the two numbers are written in.

A useful check on any whole-number-by-fraction multiplication: the denominator in the answer never changes from the original fraction's denominator (before any simplifying), because you're only ever repeating that same-sized fractional piece — just a different number of times.

When both numbers being multiplied are fractions, a unit square is a helpful way to see what's happening. To find 3/4 × 2/5, draw a square and split it into 4 equal columns (for the quarters) and 5 equal rows (for the fifths), creating 4 × 5 = 20 small equal cells in total. Shading 3 of the 4 columns and 2 of the 5 rows, the overlapping shaded region covers 3 × 2 = 6 of the 20 cells — so 3/4 × 2/5 = 6/20, which simplifies to 3/10.

This visual method generalises into a simple rule, often called Brahmagupta's rule after the Indian mathematician who first stated it in this general form in his book Brahmasphutasiddhanta, written in 628 CE: to multiply two fractions, multiply their numerators together to get the new numerator, and multiply their denominators together to get the new denominator. In symbols, a/b × c/d = (a×c)/(b×d). For example, 5/6 × 3/10 = (5×3)/(6×10) = 15/60, which simplifies to 1/4.

Multiplying numerators and denominators first, then simplifying afterward, works but can leave large numbers to simplify. It is usually faster to cancel common factors before multiplying. For 7/8 × 4/21: notice that 4 and 8 share a common factor of 4, and 7 and 21 share a common factor of 7. Cancelling these first: 7/8 × 4/21 = 1/2 × 1/3 = 1/6 — the same answer, reached with much smaller numbers.

This rule also works when one of the numbers is a whole number, simply by writing that whole number as a fraction with denominator 1. For instance, 4 × 2/7 can be written as 4/1 × 2/7 = 8/7, matching the repeated-addition method from the previous section.

With whole numbers greater than 1, multiplying always makes the result bigger: 3 × 5 = 15, which is larger than both 3 and 5. It's tempting to assume this always happens, but fractions between 0 and 1 break that assumption completely.

Multiply two numbers that are both between 0 and 1, and the product is smaller than both of them. Take 3/4 × 5/6 = 15/24 = 5/8. Comparing: 5/8 is less than 3/4, and 5/8 is also less than 5/6. This makes sense once you think about what multiplying by a fraction between 0 and 1 actually means — it means taking only a part of something, which shrinks it rather than growing it.

Multiply two numbers that are both greater than 1, and the product is bigger than both — exactly like with whole numbers. For example, 3/2 × 5/4 = 15/8 = 1 7/8, which is indeed bigger than both 3/2 (=1.5) and 5/4 (=1.25).

The most interesting case is when one number is between 0 and 1 and the other is greater than 1. Here, the product lands strictly between the two original numbers. Take 4/5 × 7/6 = 28/30 = 14/15. Checking: 4/5 = 0.8 and 7/6 ≈ 1.1667, and 14/15 ≈ 0.9333 sits right between them. Multiplying by the fraction less than 1 shrinks the larger number down, but not all the way below the smaller number.

Division can always be rewritten as a multiplication question in disguise. Asking "What is 12 ÷ 4?" is the same as asking "What number, multiplied by 4, gives 12?" — and the same trick works for fractions. Asking "What is 2/3 ÷ 5/6?" is the same as asking "What number, multiplied by 5/6, gives 2/3?"

The number that undoes a multiplication by a fraction is called its reciprocal — found simply by swapping the numerator and denominator. The reciprocal of 5/6 is 6/5, and multiplying any fraction by its own reciprocal always gives exactly 1 (since 5/6 × 6/5 = 30/30 = 1). This is the key fact that makes fraction division possible: dividing by a fraction is the same as multiplying by its reciprocal.

So the rule for dividing two fractions is: flip the divisor (find its reciprocal), then multiply. In symbols, a/b ÷ c/d = a/b × d/c = (a×d)/(b×c). Working out 2/3 ÷ 5/6 this way: flip 5/6 to get 6/5, then multiply: 2/3 × 6/5 = 12/15, which simplifies to 4/5. As another example, 3/4 ÷ 9/8 = 3/4 × 8/9 = 24/36, which simplifies to 2/3.

Just as with multiplication, there is a pattern connecting the divisor's size to how the quotient compares with the dividend. Dividing by a number between 0 and 1 always gives a quotient bigger than the dividend — dividing 6 by 1/4 gives 24, far bigger than 6, because "how many quarters fit into 6" is naturally a much bigger count than 6 itself. Dividing by a number greater than 1, on the other hand, always shrinks the dividend — dividing 5/8 by 15 gives 1/24, much smaller than 5/8.

Fraction arithmetic becomes genuinely useful once it's applied to real situations. Suppose a water tank fills at a rate of 7/10 of the tank every hour, and the tap is left open for only 2/3 of an hour. The fraction of the tank filled is found by multiplying the rate by the time: 7/10 × 2/3 = 14/30, which simplifies to 7/15 of the tank.

Fraction division shows up just as often. A tailor has 9/10 metre of fabric, and each shirt collar needs 3/10 metre. The number of collars that can be cut is found by dividing the total fabric by the amount needed per collar: 9/10 ÷ 3/10 = 9/10 × 10/3 = 90/30 = exactly 3 collars.

A car that runs 16 km on a single litre of petrol can travel 16 × 2 3/4 km on 2 3/4 litres — converting the mixed number first (2 3/4 = 11/4), this gives 16 × 11/4 = 176/4 = 44 km.

These same rules — multiply numerators and denominators together for multiplication, flip-and-multiply for division — were first stated in exactly this general form by the Indian mathematician Brahmagupta in 628 CE, and were further clarified a few centuries later by Bhaskara II, who explicitly described division as "multiplication by the reciprocal." The ideas travelled from India through the Arab world and reached Europe only around the 17th century — making fractions, in their modern form, a genuinely Indian mathematical contribution that the rest of the world adopted centuries later.