Class 6 Mathematics · Chapter 7 NotesFractions

When one whole unit is divided into several equal parts, each part is called a fractional unit or unit fraction. Sharing one roti equally between two children gives each child one half; sharing the same roti among four children gives each child one fourth. The more people share the same whole, the smaller each share becomes, so one half is greater than one fourth. A common mistake is to think that one ninth is greater than one fifth just because 9 is bigger than 5. Think of shares instead: sharing with more people means getting less, so one ninth is smaller than one fifth. Fractional units include one half, one third, one fourth, one fifth and so on, written as 1/2, 1/3, 1/4, 1/5.

• Fractional unit (unit fraction)one equal part of a whole, with numerator 1
• More equal parts of the same whole means each part is smaller
• 1/2 > 1/4, and 1/5 > 1/9, because fewer shares means a bigger share

A fraction such as three fourths means three copies of the fractional unit one fourth. If a chikki is broken into four equal pieces and a bigger piece contains three of them, that piece is 3/4 of the chikki. Dividing the same whole into six equal parts in different ways gives pieces of different shapes, but every piece is still 1/6 of the whole and all of them are of the same size. Fractional quantities can be measured by collecting fractional units together. Folding a paper strip into two, then four, then eight equal parts shows this clearly: two times 1/4 is 2/4, three times 1/4 is 3/4, and four times 1/4 makes the whole strip, that is 4/4 = 1. In a fraction such as 5/6, 5 is the numerator and 6 is the denominator.

• 3/4 means 3 times the fractional unit 1/4
• Equal parts of a whole may look different in shape but are the same size
• Numeratorthe top number, telling how many fractional units are taken
• Denominatorthe bottom number, telling the fractional unit used

Fractions can be shown as lengths on the number line. The distance between 0 and 1 is one unit long. If that distance is divided into two equal parts, each part is 1/2 unit long, so the point at the first mark is 1/2 and the next mark is 2/2, which is 1. Dividing the same unit into three equal parts gives marks at 1/3, 2/3 and 3/3. Dividing it into five equal parts gives 1/5, 2/5, 3/5, 4/5 and 5/5. Lengths less than one unit lie between 0 and 1, while lengths greater than one unit lie beyond 1. Between 0 and 1 there are countless fractions, because a unit can be divided into as many equal parts as we wish.

• Each equal part of the unit on the number line is one fractional unit
• Points to the right of 1 represent fractions greater than 1
• There are uncountably many fractions between 0 and 1

In a fraction less than 1, the numerator is smaller than the denominator. In a fraction greater than 1, the numerator is larger than the denominator, for example 3/2, 5/2 and 7/2. Such fractions can be split into whole units and a leftover part: 3/2 = 1 + 1/2, and 5/2 = 2 + 1/2. A mixed number, or mixed fraction, contains a whole number part and a fractional part that is less than 1, written as 1 1/2, 2 1/2 and so on. To write a mixed number as a single fraction, count all the fractional units: 3 3/4 means three whole units plus three fourths, which is 15/4. Not every fraction greater than 1 gives a mixed number with a fractional part, because some, like 8/4, are whole numbers.

• Numerator greater than denominator means the fraction is more than 1
• Mixed number = whole part + fractional part less than 1
• 3/2 = 1 1/2 and 5/2 = 2 1/2
• 3 3/4 written as a single fraction is 15/4

Fractions that represent the same share or the same length are called equivalent fractions. A fraction wall made of paper strips shows that 1/2 = 2/4 = 4/8, and that 1/2 = 2/4 = 3/6. Equivalent fractions also appear in sharing: one roti shared between two children, two rotis shared among four children, and three rotis shared among six children all give each child the same share, so 1/2 = 2/4 = 3/6. Multiplying or dividing both the numerator and the denominator by the same number gives an equivalent fraction. A fraction is in lowest terms, or simplest form, when its numerator and denominator have no common factor other than 1. For example, 16/20 reduces to 4/5 by dividing both numbers by 4, and 36/60 reduces to 3/5.

• Equivalent fractionsdifferent numerators and denominators, same value
• Multiply or divide numerator and denominator by the same number
• Lowest termsnumerator and denominator share no common factor except 1
• 16/20 in lowest terms is 4/5; 36/60 in lowest terms is 3/5

Comparing fractions is easy when the fractional units are the same: with equal denominators, the fraction with the larger numerator is greater. When the denominators differ, first change the fractions into equivalent fractions with the same denominator, then compare the numerators. To compare 4/5 and 7/9, use 45 as a common denominator: 4/5 = 36/45 and 7/9 = 35/45, and since 36 is greater than 35, 4/5 is greater than 7/9. A common denominator can be found by taking the product of the two denominators or a common multiple of them. Fractions can then be arranged in ascending order, from smallest to greatest, or in descending order, from greatest to smallest.

• Same denominatorcompare the numerators directly
• Different denominatorsconvert to equivalent fractions with a common denominator first
• 4/5 = 36/45 and 7/9 = 35/45, so 4/5 > 7/9

Adding fractions with the same denominator means adding the number of fractional units: 2/5 + 1/5 = 3/5, and 4/7 + 6/7 = 10/7, which can also be written as 1 3/7. When the denominators differ, first convert the fractions into equivalent fractions with the same fractional unit, then add the numerators and keep the common denominator. For example, 1/4 + 1/3 = 3/12 + 4/12 = 7/12, and 2/3 + 1/5 = 10/15 + 3/15 = 13/15. Subtraction follows the same steps: 6/7 − 4/7 = 2/7, and 3/4 − 2/3 = 9/12 − 8/12 = 1/12. If the result can be simplified, express it in lowest terms. These steps are known as Brahmagupta's method, described by him in 628 CE.

• Same denominatoradd or subtract only the numerators
• Different denominatorsmake the fractional units the same first
• 1/4 + 1/3 = 3/12 + 4/12 = 7/12
• 3/4 − 2/3 = 9/12 − 8/12 = 1/12
• Simplify the answer to lowest terms if needed

In ancient India a fraction was called bhinna in Sanskrit, meaning broken; it was also called bhaga or ansha, meaning part or piece. The way fractions are written today originated in India. In the Bakshali manuscript, from around 300 CE, one half was written as 1 over 2, very close to the modern form. This way of writing and working with fractions continued through the centuries in the work of Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE) and Mahaviracharya (c. 850 CE). The line between the numerator and denominator was introduced later by the Moroccan mathematician Al-Hassar in the 12th century, and the notation then spread to Europe and around the world. General fractions and their rules of arithmetic were first introduced in India, and Brahmagupta formally described the rules for adding and subtracting fractions that are still used today.

• Bhinna means broken; bhaga and ansha mean part or piece
• Bakshali manuscript (around 300 CE) shows fractions written much as we write them
• The fraction bar was introduced by Al-Hassar in the 12th century
• Brahmagupta described general rules for adding and subtracting fractions in 628 CE