#### Fractions

Types of Fractions :
1. Proper fraction: is a fraction that represents a part of a whole. Its numerator is less than the denominator. For example, 3/4 , 2/9 etc
2. Improper fraction: is a combination of whole and a proper fraction. For example, 7/2, 9/8 etc.
3. Mixed fraction: improper fraction can be written in another way which is called mixed fraction. For example: The improper fraction 7/4 can be written as 1 3⁄4.
4. Like fraction: is the fraction having same denominator but different numerator. For example, 3/13, 9/13, 7/13.
5. Unlike fraction: is the fraction having different denominator. For example, 6/2, 4/9, 5/4.

Refer video lesson.1 for more clarity on these topics shown above.

How to multiply fractions :
1. To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same.
For eg, 2 × 5/3 = (2×5)/3 = 10/3
2. To multiply a mixed fraction to a whole number, first convert the mixed fraction to an improper fraction and then multiply.
For eg, 3 × 2 3⁄4 = (3×11)/4 = 33/4

Fraction as an Operator 'of' :
The ‘of’ operator is used to denote the part of something. It can be replaced with a multiplication sign as shown.
For eg, 1⁄2 of 16 = 1⁄2 × 16 = 8.

Multiplication of a Fraction by a Fraction:
1. 1. we multiply two fractions as = (Product of Numerators) /(Product of Denominators).
2. 2. When two proper fractions are multiplied, the product is less than both the fractions. Or, we say the value of the product of two proper fractions is smaller than each of the two fractions.

Our Observations:
1. 1. The product of two improper fractions is greater than each of the two fractions.
2. 2. the value of the product of two improper fractions is more than each of the two fractions.
3. Check Ncert book Class-7 for more examples.

DIVISION OF FRACTIONS:
1. 1. Division of Whole Number by a Fraction is done by divide a whole into a number of equal parts such that each part is half of the whole.
For example, 1 ÷ 1/2 = 1 × 2/1
# Reciprocal of a fraction: The non-zero numbers whose product with each other is 1, are called the reciprocals of each other. For example, reciprocal of 5/9 is 9/5 and the reciprocal of 9/5 is 5/9.
2. Division of a Fraction by a Whole Number: While dividing mixed fractions by whole numbers, convert the mixed fractions into improper fractions.
3. Division of a Fraction by Another Fraction: For example, To 1/3 ÷ 5/6 =1/3 × reciprocal of 5/6 =1/3 × 6/5 = 2/5.

Refer to video lecture 2 for fraction number operations.

#### Decimals

MULTIPLICATION OF DECIMAL NUMBERS:
For example, while finding 0.1 × 0.1 and 0.2 × 0.3, you might have noticed that first we multiplied them as whole numbers ignoring the decimal point. In 0.1 × 0.1, we found 01 × 01 or 1 × 1. Similarly in 0.2 × 0.3 we found 02 × 03 or 2 × 3.
Also remember that, we counted the number of digits starting from the rightmost digit and moved towards left. We then put the decimal point there.

Multiplication of Decimal Numbers by 10, 100 and 1000:
when a decimal number is multiplied by 10, 100 or 1000, the digits in the product the are same as in the decimal number but the decimal point in the product is shifted to the right by as , many of places as there are zeros over one.

DIVISION OF DECIMAL NUMBERS:
1. Division by 10, 100 and 1000:
While dividing a number by 10, 100 or 1000, the digits of the number and the quotient are same but the decimal point in the quotient shifts to the left by as many places as there are zeros over one. For example, 2.38 ÷ 10 = 0.238, 2.38 ÷ 100 = 0.0238, 2.38 ÷ 1000 = 0.00238. Refer to the video lecture. 3 for graphical explanation.

2. Division of a Decimal Number by a Whole Number:
Let us see an example from Class 7 ncert book Pg-53, To find 19.5 ÷ 5, first find 195 ÷5. We get 39. There is one digit to the right of the decimal point in 19.5. Place the decimal point in 39 such that there would be one digit to its right. You will get 3.9.

3. Division of a Decimal Number by another Decimal Number :
Let us see an example from Class 7 ncert book Pg-54 While dividing two decimal numbers, first shift the decimal point to the right by equal number of places in both, to convert the divisor to a whole number. Then divide. Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.

For multiplication and division of decimal number refer video lecture 4.