**Notes of chapter: Fractions and Decimals are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 7.**

**(1) Fraction-**

A **fraction** means a part of a whole thing. A ball, an apple, a pizza all are one whole thing. If we cut ball in to two sections or more, every section is known as a part. If, one section or part of the ball will be coloured in red .Then, this red part is only one part of the whole and it is called as fraction of the whole (ball). In daily use language, we can say that a ball is red coloured.

** **

**(3) Types of fraction –**

**(i)** **A proper fraction** is a fraction in which numerator is smaller than its denominator.

**(ii)** An **improper fraction** is a fraction in which numerator is greater than its denominator.

**(iii)Mixed fraction** is a combination of a whole number and a proper fraction.

**Steps to Change Mixed Fraction into Improper Fraction **

**Steps to Change Improper Fraction into Mixed Fraction **

**(iv)Equivalent **fractions represent same portion of a whole.

**Method to Find Equivalent Fraction**

**(4) Operations of Fractions**–

Fractions also show all mathematical operation (+, -, × and ÷)

**(i) Addition of Fractions –**

**(a) If denominators are same**

Then add numerators with same denominator.

**Example –**

Add given fractional number.

**(b) If denominators are not same**

**Solution**

**Step 1:-**

Denominator of all given fraction are not same. So, first we will have to make denominators of al fraction same [Equivalent fraction].

Take LCM of denominators.

Numbers of denominators are 5, 7, 3.

LCM of 5, 7, 3 = 105

**(ii) Subtraction of Fractions-**

**(a) If denominators are same**

**Then subtract numerators with same denominator.**

**Example** –

**Solution**

Denominator of all given fractions are same. But numbers of numerators are different. So, first we will subtract numerators above the fraction bar and write denominator below the fraction bar.

**(b) If denominators are not same**

**Eg:-**

** **

**Solution**

**Step 1:-**

Denominator of all given fraction are not same. So, first we will have to make denominators of al fraction same [Equivalent fraction].

Take LCM of denominators.

Numbers of denominators are 5, 7.

LCM of 5, 7 = 35

**(iii) Multiplication of fraction-**

**(a) Multiplication of a fraction by a whole number**

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 same.

**Eg:-**

**(b) Multiplication of a fraction by a fraction**

When we do multiplication of two fraction, numerator and denominator of one fraction is multiplied to numerator and denominator of other fraction respectively.

**Eg:-**

**Value of the products**

**(a)** The value of the product of two proper fractions is smaller than each of two fractions.

**(b)**The value of the product of two improper fractions is more than each of the two fractions.

**(c)** The value of product of one improper and one proper fraction is less than improper fraction and greater than the proper fraction involved in the multiplication.

**(iv) Division of Fraction**

**(a)Division of whole number by a fraction**

**(b)** **Division of a fraction by a whole number**

**(c)** **Division of a fraction by another fraction**

**(5) Decimal numbers-**

**(i) **A **decimal number** is a number which has two parts, ie, whole number part and fractional part. These parts are separated by point notation. This point notation is known as **decimal**. The numbers which have decimals are called **decimals numbers**.

**Eg:-** 25.56 is a decimal number

**(ii)** Whole number part comes before decimal or at left side of decimal. Fractional part or decimal part comes after decimal or at right side of the decimal.

**(iii)Place value of the decimal numbers**

Whole numbers presents ones, tens, hundreds, thousands etc position or places of digits. Decimal numbers presents one – tenths, one – hundredths, one -thousandths etc positions or places of digits. One tenths position is just after the decimal point, one – hundredths come after it and this process is followed by other positions also.

**Eg:-** 125.567 is a decimal number

125 is whole number part.

.567 is fractional part or decimal part

**(iv)Reading of decimal numbers**

Decimal is pronounced as point. Number after decimal reads separately.

**Eg:-** 56.234

To read this number we first pronounced whole number part as a number,ie, fifty six in our example. Then we will pronounce decimal as point.

Then we read numbers after decimal as two, three, four.

Therefore, we will pronounce 56.234 as fifty six point two three four.

**Table Showing Reading of a Decimal Number**

S.N. |
Decimal numbers |
Reading of decimal number |

1. | 76.678 | Seventy six point six, seven, eight |

2. | 124.098 | One hundred twenty four point zero, nine, eight |

3. | 1098.980 | One thousand ninety eight point nine, eight, zero |

**Examples-**

**(v) Comparing of the decimals numbers-**

For comparing numbers first compare numbers on the left side of decimal and then compare numbers on the right side of decimals. To compare number on the right side first compare number at tenths place than at hundredths place and continue this process till we get answer.

**Eg:-** Which is greater 14.24 or 14.34

Comparing numbers on the left side of the decimals

14=14

Comparing numbers on the right side of the decimals tenths place

2<3

Hence 14.34>14.24

**Eg.:-**Which is greater 4.04 or 4.23

Comparing numbers on the left side of the decimals

4=4

Comparing numbers on the right side of the decimals tenths place

0<2

Hence 4.04<4.23

**(vi)Addition of the decimals Numbers-**

Procedure of addition of decimals numbers is similar to addition of whole numbers.

**(a)Addition of the decimal numbers without carryover-**

**Step 1**

First add numbers at fractional part or at right side of decimals.

**Step 2**

Now, add numbers of whole numbers part.

**Eg:-** Add 23. 56 and 34.12

**(b) Addition of the decimal numbers with carryover-**

**Step 1**

First add numbers at fractional part or at right side of decimals.

**Step 2**

Now, add numbers of whole numbers part .Add carryover of fractional part also with the sum of ones of whole number part.

**Eg:-** Add 56.234 and 45.782

**(vii)Subtraction of decimal numbers-**

Procedure of subtraction of decimals numbers is similar to subtraction of whole numbers.

**(a)Subtraction of the decimal numbers without borrowing-**

**Step 1:**

First subtract numbers at fractional part or at right side of decimals.

**Step 2**

Now, subtract the numbers of whole numbers part.

** Eg:-** Subtract 34.01 from 67.09

Ans-

**(b)Subtraction of the decimal numbers with borrowing-**

**Step 1:**

First subtract numbers at fractional part or at right side of decimals.

**Step 2**

Now, subtract the numbers of whole numbers part .Borrow numbers from fractional par.

**Eg:-** Subtract 45.67 from 87.02

Ans-

**(viii)Multiplication of decimal numbers-**

**(a) Multiplication of decimal numbers by whole number-**

**Step 1**

Multiply whole number digits without putting decimal.

**Step 2**

Count digits of decimal number form right of decimal [Fractional Part].

**Step 3**

Count multiplication digits from ones and put decimal where it equals with counted digits of fractional part.

**Eg:-** Multiply 2.03 by 2

Ans-

**(b) Multiplication of decimal numbers by decimal numbers-**

**Step 1**

Multiply whole number digits without putting decimal.

**Step 2**

Count digits of both numbers together of fractional part.

**Step 3**

Count multiplication digits from ones and put decimal where it equals with counted digits of fractional part.

**Ans-**

**Ans-**

**(c)Multiplication of decimal Numbers by 10, 100, 1000-**

**Step 1**

Product of digits is same as in the decimal number.

**Step 2**

Decimal point in the product is shifted to the right by as, many of places as there is zero over one.

**Eg-**

**(ix)Division of decimal numbers-**

**(a)Division by 10,100 and 1000-**

**Step 1**

The digits of the number and quotient are same.

**Step 2**

The decimal point will be place where number of zero equals the places of digits from extreme right.

**Eg:-**

**(b)Division by any whole number-**

**(A) If remainder is zero**

**Eg:-** 24.2 is divided by 2

**Step1**

Divide whole part of the number by 2.

**Step 2**

Put point in quotient.

**Step 3**

Divide fractional part as we do in whole numbers.

**(b) If remainder is not zero**

**Eg:-** Divide 54.67 by 12

**Ans-**

**Step 1**

Divide whole part of the number by 12.

**Step 2**

Put point in quotient.

**Step 3 –**

Divide fractional part as we do in whole numbers. If any digit remains in remainder other than zero, use zero with remainder at ones place and divide it with divisor. We can take zero after putting point in quotient.

We can continue division by adding zero at ones place of remainder but no need to do division more than three places until mentioned in question.

**(iii)Division of decimal numbers by decimals numbers-**

**Eg:-** Divide 64.02 by 12.32

**Ans-**

**Step 1**

To remove decimals count digits after decimals if they are same just remove decimal and divide following same processor of division of whole numbers.

**We can solve it by two methods**

**Method 1**

**Step 1**

Divide as in whole numbers.

**Step 2**

Put decimal in quotient if get remainder.

**Step 3**

After putting point or decimal in quotient take zero with remainder at one place and divide.

Repeat this process up to three point of decimal.

**Method 2**

**Step 1**

First reduce it to lowest form and then divide as in whole numbers

**Step 2**

Divide 1603 by 308

**Helping Topics**