In mathematics, a fraction is used to represent a portion of something larger. It represents the whole's equal pieces. A fraction's numerator and denominator are the two elements. The numerator is the number at the top, and the denominator is the number at the bottom.The denominator defines the total number of equal parts in the total, whereas the numerator indicates how many equal parts were taken. 5/10, for example, is a fraction. The denominator is 10 and the numerator is 5.

Two integers are placed on top of each other and separated by a line to represent a fraction. The number on top is the numerator, and the number below is the denominator. Example: 34, which denotes three parts from four equal divisions.

To represent a fraction on a number line, we divide the line segment between two whole numbers into n equal segments, where n is the denominator.

For example, to represent 1/5 or 3/5, we divide the line into 5 equal sections between 0 and 1. The numerator then indicates the number of divisions to be marked.

Example 1: 7×(1/3) = 7/3

Example 2: 5 ×(7/45) = 35/45; we get 7/9 by dividing the numerator and denominator by 5.

 

The product of numerators/product of denominators is the basic formula for multiplying fractions by fractions.

 

Example 1: (3/5) × (12/13) = 36/65

Example 2 : Multiplication of mixed fractions

4(⅔) × 1(1/7) 

To multiply improper fractions, first convert mixed fractions to improper fractions.

14/3 × 8/7

 

Multiplication is implied by the 'of' operator.

For instance, 1/6 of 18 is 3

Alternatively, 1/2 of 11 Equals (1/2) 11 = 11/2

A Fraction's Reciprocal

When the numerator and denominator of a fraction are swapped, the fraction becomes reciprocal.

1/n is the reciprocal of any number n.

For example, the reciprocal of 3/5 is 5/3.

Despite the fact that zero divided by any number equals zero, we can't identify reciprocals because a number divided by 0 is undefined.

Example : Reciprocal of 0/7 ≠ 7/0

Division of a whole number by a fraction is done by multiplying the entire number by the fraction's reciprocal.

Example: 64÷(7/5) = 64×(5/7) = 45.7

 

We multiply the fraction with the reciprocal of the whole number to divide a fraction by a whole number.

Example: (8/6)÷4 = (8/6)×(1/4) = 3

 

Multiply the dividend by the reciprocal of the divisor to divide a fraction by another fraction.

Example: (2/7) ÷ (5/7) = (2/7) × (7/5) = 2/5

Proper fractions denote a portion of a larger total. The numerator is larger than the denominator. For instance, 1/4, 7/9, and 50/51. Proper fractions are those that are greater than 0 and smaller than 1.

 

The numerator of an improper fraction is more than or equal to the denominator.

Example: 45/6, 6/5. Fractions that are higher than or equal to one are called improper fractions.

 

A mixed fraction is one in which a whole number and a proper fraction are combined.

Example: 43/5 can be written as 8(⅗)

 

Fraction conversion: A mixed fraction can be expressed as an improper fraction, and an incorrect fraction can be represented as a mixed fraction. A fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number) are known as improper fraction. Examples of Improper fractions are 5/3 (five thirds) and 9/8 (nine eighths).

 

Like fractions: Fractions with the same denominator are referred to as similar fractions. For instance, 5/7 and 3/7. We can compare them as follows: (5/7) > (3/7)

 

Unlike fractions: Fractions having different denominators are referred to as unlike fractions. 5/3, 9/2 are two examples. To compare them, we use the denominator's L.C.M.

 

Here the L.C.M is 6 So, (5/3)×(2/2) , (9/2)×(3/3)

⇒ 10/6, 27/6

⇒ 27/6 > 10/7

 

 

Introduction: Decimal

Smaller numbers are represented using decimal numbers, which are smaller than the unit 1. Since each place value is expressed by a power of ten, the decimal number system is also known as the base 10 system.

There are two parts to a decimal number:

(i) the integral component (before the decimal point) and 

(ii) the fractional part (after the decimal point).

A decimal separator(.) known as the decimal point separates these two.

 

For example, a decimal number is written as 566.8 or 25.97.

 

The order of ten increases the numbers to the left of the decimal point, while the order of ten decreases the numbers to the right of the point.

The number 566.8 can be read as 'five hundred and sixty-six and eight tenths' in the example above.

⇒5×100 + 6×10 + 6×1 + 8×(1/10)

 

A decimal number can be written as a fraction, and vice versa. Example 1.5 = 15/10 = 3/2 or 3/2 = 1.5 = 15/10 = 3/2

Multiply decimal numbers with whole numbers as though they were entire numbers. After the decimal point, the product will have the same amount of digits as the decimal number.

For example, 11.3  × 5 = 56.5

 

When a decimal is multiplied by a power of ten, the decimal point shifts to the right by the number of zeros in the power.

For example, 45.678 ×10 = 456.78 (decimal point shifts one place to the right) or 45.678 ×1000 = 45678 (decimal point shifts by 3 places to the right)

Multiply the decimal numbers without decimal points, then add a decimal point to the answer in the same number of places as the total number of places right to the decimal points in both numbers.

Taking a decimal number and dividing it by a whole number:

45.35/5 is an example.

To convert a decimal number to a fraction, follow these steps: 45.35 is the same as 4535/100.

Step 2: Take the fraction and divide it by the whole number: (4535/100)÷5 = (4535/100) × (1/5) = 9.07

Example 1: 45.25/0.5

Step 1: Fractionally convert both decimal numbers: 45.25 = 4525/100 and 0.5 = 5/10

Step 2: Subtract the fractions from the total: (4525/100)÷(5/10) = (4525/100)×(10/5) = 90.5

When a decimal number is divided by a power of ten, the decimal point shifts to the left by the number of zeros in the power of ten.

Example: 98.765÷100=0.98765 Infinity

 

When the denominator of a fraction is extremely small (nearly zero), the value of the fraction tends to infinity.

For example, 999999/0.000001 = 999999000001, which is a very huge number.

• • Division of a whole number by a fraction

Consider this example, 54 ÷ (6/9). In this case, the concept of reciprocation comes into the picture. In reciprocation, the reciprocal of the original number of the fraction is obtained when 1 is divided by that number or fraction. So, if you need to find the reciprocal of 2/5, then it is interpreted as 1 ÷ (2/5), and the final result becomes (5/2). So, using this, the original problem of division of a whole number by a fraction, 54 ÷ (6/9), can be written as 54 x (9/6), and the result becomes (54 x 9) / 6 = 81.

• • Division of a fraction by a whole number

Under this, a fraction is to be divided by a whole number. You can utilise reciprocals to multiply the fraction by the reciprocal of the entire number once more. To better understand how to divide a fraction by a whole number, try this example: (2/5) (9) can be written as (2/5) x (1/9), and since you already know how to solve it, the final result is 2/45.

Exercise 2.1 Page: 31

1. Solve:

(i) 2 – (3/5)

Solution:-

To subtract two unlike fractions, convert them to like fractions first.

LCM of 1, 5 = 5

Let us now convert each of the given fractions to an equivalent fraction with 5 as the denominator.

= [(2/1) × (5/5)] = (10/5)

= [(3/5) × (1/1)] = (3/5)

Now,

= (10/5) – (3/5)

= [(10 – 3)/5]

= (7/5)

(ii) 4 + (7/8)

Solution:-

To add two unlike fractions, we first need to change them to the like fractions.

LCM of 1, 8 = 8

Now, convert each of the fractions into equivalent fractions with 8 as the denominator.

= [(4/1) × (8/8)] = (32/8)

= [(7/8) × (1/1)] = (7/8)

Now,

= (32/8) + (7/8)

= [(32 + 7)/8]

= (39/8)

=4 ⅞ 

(iii) (3/5) + (2/7)

Solution:-

To add two unlike fractions, let us change them to like fractions first.

LCM of 5, 7 = 35

Now, convert each of the fractions into equivalent fractions with denominator 35.

= [(3/5) × (7/7)] = (21/35)

= [(2/7) × (5/5)] = (10/35)

Now,

= (21/35) + (10/35)

= [(21 + 10)/35]

= (31/35)

(iv) (9/11) – (4/15)

Solution:-

To subtract two unlike fractions, change them to like fractions first.

LCM of 11 and 15 = 165

Now, change the given fractions into equivalent fractions with denominator = 165.

= [(9/11) × (15/15)] = (135/165)

= [(4/15) × (11/11)] = (44/165)

Now,

= (135/165) – (44/165)

= [(135 – 44)/165]

= (91/165)

(v) (7/10) + (2/5) + (3/2)

Solution:- To combine two unlike fractions, convert them to like fractions first.

LCM of 10, 5, 2 = 10

Let us now convert each of the given fractions to an equivalent fraction with the denominator as 35.

= [(7/10) × (1/1)] = (7/10)

= [(2/5) × (2/2)] = (4/10)

= [(3/2) × (5/5)] = (15/10)

Now,

= (7/10) + (4/10) + (15/10)

= [(7 + 4 + 15)/10]

= (26/10)

= (13/5)

=2 ⅗ 

(vi)  2 ⅔ + 3 ½ 

Solution:-

To begin, transform the mixed fraction to an improper fraction.

= 2 ⅔ = 8/3

= 3 ½ = 7/2

To combine two unlike fractions, convert them to like fractions first.

LCM of 3, 2 = 6

Let us now convert each of the given fractions to an equivalent fraction with the denominator as 6.

= [(8/3) × (2/2)] = (16/6)

= [(7/2) × (3/3)] = (21/6)

Now,

= (16/6) + (21/6)

= [(16 + 21)/6]

= (37/6)

= 6 ⅙

2. Arrange the following in descending order:

(i) 2/9, 2/3, 8/21

Solution:-

LCM of 9, 3, 21 = 63

Let us now convert each of the given fractions to an equivalent fraction with the denominator 63.

[(2/9) × (7/7)] = (14/63)

[(2/3) × (21/21)] = (42/63)

[(8/21) × (3/3)] = (24/63)

Clearly,

(42/63) > (24/63) > (14/63)

Hence,

(2/3) > (8/21) > (2/9)

As a result, the given fractions are(2/3),(8/21), and(2/9) in descending order.

(ii) 1/5, 3/7, 7/10

Solution:-

LCM of 5, 7, 10 = 70

Let us now convert each of the given fractions to an equivalent fraction with a denominator of 70.

[(1/5) × (14/14)] = (14/70)

[(3/7) × (10/10)] = (30/70)

[(7/10) × (7/7)] = (49/70)

Clearly,

(49/70) > (30/70) > (14/70)

Hence,

(7/10) > (3/7) > (1/5)

As a result, the given fractions are(7/10),(3/7), and(1/5) in descending order.

3. A "magic square" is one in which the sum of the integers in each row, column, and diagonal is the same. Is this a magic square?

 

4/11	9/11	2/11
3/11	5/11	7/11
8/11	1/11	6/11