What is a fraction?

Fractions as equal shares

Share one roti equally between 2 children and each gets 1/2 (one half) roti. Share the same roti among 4 children and each gets 1/4 (one quarter). The second share is smaller, because more children are sharing the same roti, so 1/2 > 1/4.

So a fraction answers a simple question: when a whole is shared equally, how much is each share?

Fractional units

When one whole is divided into equal parts, each part is called a fractional unit, or a unit fraction: 1/2, 1/3, 1/4, 1/5, … , 1/10, … , 1/100.

Which is bigger, 1/5 or 1/9? Many students say 1/9, because 9 is bigger than 5. Think of shares instead. One roti shared by 5 children gives a bigger piece than one roti shared by 9 children. So 1/9 < 1/5, and in the same way 1/200 < 1/100.

Parts of a whole

A whole chikki can be broken into equal pieces in many ways. Cut it into 6 equal pieces in two different ways and the pieces may have different shapes, but each one is still 1/6 of the chikki, because each is one of 6 equal parts. What matters is that the parts are equal, not that they look the same.

Measuring with fractional units

Take a paper strip and call its length 1 unit. Fold it in half: each part is 1/2. Fold it in half again: there are 4 equal parts, each 1/4. One more fold gives 8 parts of 1/8.

Now we can measure using these units. Three parts of 1/4 make 3/4:

3/4 = 1/4 + 1/4 + 1/4 = 3 times 1/4

We usually read 3/4 as “three quarters” or “three upon four”. Reading it as “3 times 1/4” is very useful, because it shows the fractional unit (1/4) and how many of them there are (3). In a fraction such as 5/6, the top number 5 is the numerator and the bottom number 6 is the denominator.

Fractions on the number line

The distance from 0 to 1 on a number line is 1 unit. Divide it into 4 equal parts and each part is 1/4. Three hops of 1/4 from 0 land on 3/4. Divide it into 5 equal parts instead, and the marks are 1/5, 2/5, 3/5, 4/5.

Every fraction has its own point on the number line. How many fractions lie between 0 and 1? Endlessly many: between any two of them you can always fit another.

Fractions greater than 1: mixed fractions

Look at 1/2, 3/4 and 2/5: the numerator is smaller than the denominator, and each fraction is less than 1. In 3/2, 5/2 and 7/4 the numerator is bigger, and each is more than 1.

To see how many wholes such a fraction holds, group its fractional units into wholes:

• 7/2 = 2/2 + 2/2 + 2/2 + 1/2 = 3 + 1/2, written 3 1/2.
• 8/3 = 3/3 + 3/3 + 2/3 = 2 + 2/3, written 2 2/3 and read “two and two thirds”.

A mixed fraction (or mixed number) has a whole part and a fractional part less than 1. To go back, count the fractional units: 3 3/4 = 4/4 + 4/4 + 4/4 + 3/4 = 15/4. When the numerator is a multiple of the denominator, there is no fractional part at all: 8/4 = 2, a whole number.

Words for fractions in our languages

Fractions have been named in India since ancient times. The Rig Veda calls 3/4 tri-pada, much like “teen paav” in everyday Hindi and “mukkaal” in Tamil. Hindi also has words such as paav (1/4), aadha (1/2), sava (1 1/4), dedh (1 1/2) and dhai (2 1/2). Ask at home which words your family uses. Next, see how different fractions can stand for the same share in equivalent fractions.