Class 6 Maths Chapter 7: Fractions Notes

What is a fraction?

When some whole number of things is shared equally, a fraction tells us how much each share is. One roti shared equally by 2 children gives each child 1/2 roti; shared by 4 children, each child gets 1/4 roti. The second share is smaller, because more children share the same roti, so 1/2 > 1/4.

What are fractional units?

When one whole is divided into equal parts, each part is a fractional unit (or unit fraction): 1/2, 1/3, 1/4, … , 1/10, … , 1/100. The more parts we make, the smaller each part is, so 1/9 < 1/5 and 1/200 < 1/100. Pieces of the same fractional unit can have different shapes and still be the same size, as long as they are equal shares of the same whole. Learn more in what is a fraction.

How do we read and measure fractions?

We measure a quantity by counting fractional units. Fold a paper strip in half twice and you get 4 equal parts: 3 of them make 3/4, which means 3 times 1/4. 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 and mixed fractions

Divide the length from 0 to 1 into equal parts, and every fraction gets its own point on the number line. When the numerator is bigger than the denominator, the fraction is more than 1 and can be written as a mixed fraction: a whole part and a fractional part less than 1. For example, 7/2 = 3 1/2 and 8/3 = 2 2/3. Going back, 3 3/4 = 15/4.

What are equivalent fractions?

Fractions that stand for the same share are equivalent fractions. A fraction wall shows that 1/2 = 2/4 = 3/6 = 4/8. Sharing gives the same result: 2 cakes among 5 children and 4 cakes among 10 children both give 2/5 = 4/10. Multiplying or dividing the numerator and denominator by the same number gives an equivalent fraction. A fraction is in lowest terms when its numerator and denominator have no common factor except 1, so 16/20 in lowest terms is 4/5. See equivalent fractions.

How do we compare fractions?

With the same denominator, compare the numerators: 5/8 > 3/8. With the same numerator, the smaller denominator gives the bigger fraction: 4/7 > 4/8. Otherwise, first rewrite both fractions with the same denominator: 4/5 = 36/45 and 7/9 = 35/45, so 4/5 > 7/9. See comparing fractions.

How do we add and subtract fractions?

With the same denominator, add or subtract the numerators: 2/5 + 1/5 = 3/5 and 6/7 − 4/7 = 2/7. With different denominators, first find equivalent fractions with a common denominator: 1/4 + 1/3 = 3/12 + 4/12 = 7/12, and 3/4 − 2/3 = 9/12 − 8/12 = 1/12. This is Brahmagupta’s method (628 CE). See adding and subtracting fractions.

A pinch of history

In ancient India a fraction was called bhinna, ‘broken’, and also bhaga or ansha, ‘part’. The way the whole world writes fractions today began in India: the Bakshali manuscript, from about 300 CE, writes a half as 1 above 2. The line between the two numbers was added later by the Moroccan mathematician Al-Hassar in the 12th century. Brahmagupta’s rules for working with fractions reached Europe through the Arabs and are the rules we still use.