Monthly Archives: September 2022

When we fold a paper in such a way that the picture is divided into two equal halves then the line which divides the picture into two halves is called a Line of Symmetry.

Here the line divides the star into two halves so it is the line of symmetry. It is also called the Mirror Line because if we place the mirror on that line then one side of the picture will fall exactly on the other side of the picture.

Non-symmetrical Figure

This figure is not symmetrical as if we fold the image from the dotted line then it does not divide it into two equal halves.

Making Symmetric Figures: Ink-blot Devils

To make an ink-blot pattern-

• Take a piece of paper and fold it in half.
• Put some drops of ink on one side of the paper.
• Then press the halves together.
• It will make a symmetric pattern with the fold as the line of symmetry.

Inked-string pattern

To make an inked string pattern-

• Take a piece of paper and fold it in half.
• Dip a string in different colours and arrange it on the one side of the paper.
• Press the two halves together and pull the string.
• It will make a symmetric inked string pattern with the fold as the line of symmetry.

Two Lines of Symmetry

Some figures have two lines of symmetry.

1. A Rectangle

Take a rectangular sheet and fold it horizontally in two equal halves and then again fold it vertically in two equal halves. After opening it, we get two lines of symmetry of the rectangular sheet.

2. More Figures with two Line of Symmetry

If we take a rectangular piece of paper and double fold it to make two lines of symmetry and cut it in some new shape then after opening it we will get a new image that too with the two lines of symmetry.

Construction of figure with two Lines of Symmetry

1. To draw a figure with two lines of symmetry, take one figure.

2. Let L and M be the two lines of symmetry.

3. Draw the figure in such a way that L is the line of symmetry,

4. Now complete the figure by drawing the remaining part so that M will also become the line of symmetry.

Hence this is the final figure with two lines of symmetry.

Multiple Lines of Symmetry

Take a square sheet of paper and fold it in two halves vertically and again horizontally .open it and fold it in two equal halves diagonally then again open it and fold it along another diagonal.

When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.

Some more images with more than two lines of symmetry

• Equilateral triangle will have three lines of symmetry.
• Square will have four lines of symmetry.
• Regular pentagon will have five lines of symmetry.
• Regular hexagon will have six lines of symmetry.

Some Real-life Examples of Symmetry

In Taj Mahal and the butterfly there is one line of symmetry and there are so many other things also in our daily life which are having one or more line of symmetry.

Reflection and Symmetry

The line of symmetry is also called Mirror Line because the mirror image of an object is symmetrical to the image. When we see an object in the mirror then there is no change in the length and angles of the object except one thing i.e. the image is opposite to the original image.

Some Examples of Reflection Symmetry

1. Paper Decoration

We can use a rectangular sheet to fold and create some intricate patterns by cutting paper.

2. Kaleidoscope

In Kaleidoscope, mirrors are used to create pictures having various lines of symmetry. Two mirrors strips forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.

Example

Which alphabet will remain same after reflection symmetry? Check for R, C, N, A and T.

Solution

In the alphabet reflection symmetry, the alphabets look opposite in the mirror i.e. the alphabet written from right to left will appear as written from left to right.

Hence C, N and R will not look the same after reflection.

Hence A and T will look same after reflection symmetry.

If we compare two quantities using division then it is called ratio. It compares quantities in terms of ‘How many times’. The symbol to represent ratio is “:”.

It reads as “4 is to 3”

It can also be written as 4/3.

Example

If there are 35 boys and 25 girls in a class, then what is the ratio of

• Number of boys to total students
• Number of girls to total students.

Solution

In the ratio, we want the total number of students.

Total number of students = Number of boys + Number of girls

35 + 25 = 60

• Ratio of number of boys to total number of students

• The ratio of the number of girls to the total number of students

The unit must be same to compare two quantities

If we have to compare two quantities with different units then we need to convert them in the same unit .then only they can be compared using ratio.

Example

What is the ratio of the height of Raman and Radha if the height of Raman is 175 cm and Radha is 1.35 m?

Solution

The unit of the height of Raman and Radha is not same so convert them in the same unit.

Height of Radha is 1.35 m = 1.35 × 100 cm = 135 cm

The ratio of the height of Raman and Radha 

Equivalent Ratios

If we multiply or divide both the numerator and denominator by the same number then we get the equivalent ratio. There could be so many equivalent ratios of the same ratio.

In the case of equivalent ratios only their value changes but they represent the same portion of the quantity.

Example

Find two equivalent ratios of 2/4.

Solution

To get the equivalent ratio we multiply both the numerator and denominator with 2.

To get another equivalent ratio we divide both the numerator and denominator with 2.

From the above figure, we can see that in all the equivalent ratios only the number of equal parts is changing but all the ratios are representing the half part of the circle only.

The Lowest form of the Ratio

If there is no common factor of numerator and denominator except one then it is the lowest form of the ratio.

Example

Find the lowest form of the ratio 25: 100.

Solution

The common factor of 25 and 100 is 25, so divide both the numerator and denominator with 25.

Hence the lowest ratio of 25: 100 is 1: 4.

Proportion

If we say that two ratios are equal then it is called Proportion.

We write it as a: b : : c: d or a: b = c: d

And reads as “a is to b as c is to d”.

Example

If a man runs at a speed of 20 km in 2 hours then with the same speed would he be able to cross 40 km in 4 hours?

Solution

Here the ratio of the distances given is 20/40 = 1/2 = 1: 2

And the ratio of the time taken by them is also 2/4 = 1/2 = 1: 2

Hence the four numbers are in proportion.

We can write them in proportion as 20: 40 : : 2: 4

And reads as “20 is to 40 as 2 is to 4”.
 

Extreme Terms and Middle Terms of Proportion

The first and the fourth term in the proportion are called extreme terms and the second and third terms are called the Middle or the Mean Terms.

In this statement of proportion, the four terms which we have written in order are called the Respective Terms.

If the two ratios are not equal then these are not in proportion.

Example 1

Check whether the terms 30,99,20,66 are in proportion or not.

Solution 1.1

To check the numbers are in proportion or not we have to equate the ratios.

As both the ratios are equal so the four terms are in proportion.

30: 99 :: 20: 66

Solution 1.2

We can check with the product of extremes and the product of means.

In the respective terms 30, 99, 20, 66

30 and 66 are the extremes.

99 and 20 are the means.

To be in proportion the product of extremes must be equal to the product of means.

30 × 66 = 1980

99 × 20 = 1980

The product of extremes = product of means

Hence, these terms are in proportion.
 

Example 2

Find the ratio 30 cm to 4 m is proportion to 25 cm to 5 m or not.

Solution 2

As the unit is different so we have to convert them into the same unit.

4 m = 4 × 100 cm = 400 cm

The ratio of 30 cm to 400 cm is

5 m = 5 × 100 cm = 500 cm

Ratio of 25 cm to 500 cm is

Here the two ratios are not equal so these ratios are not in proportion.

3: 40 ≠ 1: 20

Unitary Method

If we find the value of one unit then calculate the value of the required number of units then this method is called the Unitary method.

Example 1

If the cost of 3 books is 320 Rs. then what will be the cost of 6 books?

Solution 1

Cost of 3 books = Rs. 320

Cost of 1 book = 320/3 Rs.

Cost of 6 books = (320/3) × 6 = 640 Rs.

Hence, the cost of 6 books is Rs. 640.
 

Example 2

If the cost of 20 toys is Rs. 4000 then how many toys can be purchased for Rs. 6000?

Solution 2

In Rs. 4000, the number of toys can be purchased = 20

In Rs. 1, the number of toys can be purchased = Rs. 20/4000

Therefore, in Rs. 6000, the number of toys can be purchased = (20/4000) × 6000 = 30

Hence, 30 toys can be purchased by Rs. 6000.

Introduction to Algebra

There are so many branches of mathematics-

• The study of numbers is called Arithmetic.
• The study of shapes is called Geometry.
• The study to use the letters and symbols in mathematics is called Algebra.

Algebra

Algebra is a part of mathematics in which the letter and symbols are used to represent numbers in equations. It helps us to study about unknown quantities.

Matchstick Patterns

No. of matchsticks used to make 1^st square = 4

No. of matchsticks used to make 2^nd square = 7

No. of matchsticks used to make 3^rd square = 10

So, the pattern that we observe here is 3n + 1

With this pattern, we can easily find the number of matchsticks required in any number of squares.

Example

How many matchsticks will be used in the 50^th figure?

Solution

3n + 1

3 × 50 + 1

= 151 matchsticks

The Idea of a Variable

Variable refers to the unknown quantities that can change or vary and are represented using the lowercase letter of the English alphabets.

One such example of the same is the rule that we used in the matchstick pattern

3n + 1

Here the value of n is unknown and it can vary from time to time.

More Examples of Variables

• We can use any letter as a variable, but only lowercase English alphabets.
• Numbers cannot be used for the variable as they have a fixed value.
• They can also help in solving some other problems.

Example: 1

Karan wanted to buy story books from a bookstall. She wanted to buy 3 books for herself, 2 for her brother and 4 for 2 of her friends. Each book cost Rs.15.how much money she should pay to the shopkeeper?

Solution: 1

Cost of 1 book = Rs.15

We need to find the cost of 9 books.

In the current situation, a (it’s a variable) stands for 9

Therefore,

Cost of 9 books = 15 × 9

= 135

Therefore Karan needs to pay Rs.135 to the shopkeeper of the bookstall.

The variable and constant not only multiply with each other but also can be added or subtracted, based on the situation.

Example: 2

Manu has 2 erasers more than Tanu. Form an expression for the statement.

Solution: 2.1

Erasers that Tanu have can be represented using a variable (x)

Erasers that Manu have = erasers that Tanu have + 2

Erasers with Manu = x + 2

Solution: 2.2

Erasers that Manu have can be represented using a variable (y)

Erasers that Tanu has = erasers that Manu have – 2

Erasers with Tanu = y – 2

Use of Variables in Common Rules (Geometry)

1. Perimeter of Square

The perimeter of a square = Sum of all sides

= 4 × side

= 4s

Thus, p = 4s

Here s is variable, so the perimeter changes as the value of side change.

2. Perimeter of Rectangle

Perimeter of rectangle = 2(length + breadth)

= 2 (l + b) or 2l + 2b

Thus, p + 2 × (l + b) or 2l + 2b

Where, l and b are variable and the value of perimeter changes with the change in l and b.

Use of Variables in Common Rules (Arithmetic)

1. Commutativity of Addition

5 + 4 = 9

4 + 5 = 9

Thus, 5 + 4 = 4 + 5

This is the commutative property of addition of the numbers, in which the result remains the same even if we interchanged the numbers.

a + b = b + a

Here, a and b are different variables.

Example

a = 16 and b = 20

According to commutative property

16 + 20 = 20 + 16

36 = 36

2. Commutativity of Multiplication

8 × 2 = 16

2 × 8 = 16

Thus, 8 × 2 = 2 × 8

This is the commutative property of multiplication, in which the result remains the same even if we interchange the numbers.

a × b = b × a

Here, a and b are different variables.

Example

18 ×12 = 216, 12 ×18 = 216

Thus, 18 × 12 = 12 × 18

3. Distributivity of Numbers

6 × 32

It is a complex sum but there is an easy way to solve it. It is known as the distributivity of multiplication over the addition of numbers.

6 × (30 + 2)

= 180 + 12

= 192

Thus, 6 × 32 = 192

A × (b + c) = a × b + a × c

Here, a, b and c are different variables.

4. Associativity of Addition

This property states that the result of the numbers added will remain same regardless of their grouping.

(a + b) + c = a + (b + c)

Example

(4 + 2) + 7 = 4 + (2 + 7)

6 + 7 = 4 + 9

13 = 13

Expressions

Arithmetic expressions may use numbers and all operations like addition, subtraction, multiplication and division

Example

2 + (9 – 3), (4 × 6) – 8 etc…

(4 × 6) – 8 = 24 – 8

= 16

Expressions with variable

We can make expressions using variables like

2m, 5 + t etc…..

An expression containing variable/s cannot be analyzed until its value is given.

Example

Find 3x – 12 if x = 6

Solution

(3 × 6) – 12

= 18 – 12

= 5

Thus,

3x – 12 = 5

Formation of Expressions

Practical use of Expressions

Example

3 boys go to the theatre. The cost of the ticket and popcorn is $33 and $15 respectively. What is the cost per person?

Solution

Let’s say,

 x = cost of ticket per person

 y = cost of popcorn/person

Total cost of the movie (ticket + popcorn) per person = x + y

Total cost of ticket + popcorn for 3 boys = 3(x + y)

= 3 (33 + 15)

= 3 (48)

= 144

Hence the total cost of movie ticket and popcorn for 3 boys is $144.

Equation

If we use the equal sign between two expressions then they form an equation.

An equation satisfies only for a particular value of the variable.

The equal sign says that the LHS is equal to the RHS and the value of a variable which makes them equal is the only solution of that equation.

Example

3 + 2x = 13

5m – 7 = 3

p/6 = 18

If there is the greater then or less than sign instead of the equal sign then that statement is not an equation.

Some examples which are not an equation

23 + 6x  > 8

6f – 3 < 24

The Solution of an Equation

The value of the variable which satisfies the equation is the solution to that equation. To check whether the particular value is the solution or not, we have to check that the LHS must be equal to the RHS with that value of the variable.

Trial and Error Method

To find the solution of the equation, we use the trial and error method.

Example

Find the value of x in the equation 25 – x = 15.

Solution

Here we have to check for some values which we feel can be the solution by putting the value of the variable x and check for LHS = RHS.

Let’s take x = 5

25 – 5 = 15

20 ≠ 15

So x = 5 is not the solution of that equation.

Let’s take x = 10

25 – 10 = 15

15 = 15

LHS = RHS

Hence, x = 10 is the solution of that equation.

The closed 2-D shapes are referred to as plane figures.

Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.

Perimeter

If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.

If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.

Example

Find the Perimeter of the given figure.

Solution

Perimeter = Sum of all the sides

= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m

= 68 m

The Perimeter of a Rectangle

A rectangle is a closed figure with two pairs of equal opposite sides.

Perimeter of a rectangle = Sum of all sides

= length + breadth + length + breadth

Thus, Perimeter of a rectangle = 2 × (length + breadth)

Example: 1

The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.

Solution: 

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the pool = 2 × (16 + 12)

= 2 × 28

= 56 meters

Example: 2

Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.

Solution:

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the farm = 2 × (24 + 18)

= 2 × 42

= 84 meter

Cost of fencing = 84 × 8

= Rs. 672

Thus the cost of fencing the farm is Rs. 672/-.

Regular Closed Figure

Figures with equal length of sides and an equal measure of angles are known as Regular Closed Figures or Regular Polygon.

Perimeter of Regular Polygon = Number of sides × Length of one side

Perimeter of Square

Square is a regular polygon with 4 equal sides.

Perimeter of square = side + side + side + side

Thus, Perimeter of a square = 4 × length of a side

Example

Find the perimeter of a square having side length 25 cm.

Solution

Perimeter of a square = 4 × length of a side

Perimeter of square = 4 × 25

 = 100 cm

Perimeter of an Equilateral Triangle

An equilateral triangle is a regular polygon with three equal sides and angles.

Perimeter of an equilateral triangle = 3 × length of a side

Example

Find the perimeter of a triangle having each side length 13 cm.

Solution

Perimeter of an equilateral triangle = 3 × length of a side

Perimeter of triangle = 3 × 13

= 39 cm

Perimeter of a Regular Pentagon

A regular pentagon is a polygon with 5 equal sides and angles.

Perimeter of a regular pentagon = 5 × length of one side

Example

Find the perimeter of a pentagon having side length 9 cm.

Solution

Perimeter of a regular pentagon = 5 × length of one side

Perimeter of a regular pentagon = 5 × 9

= 45 cm

Perimeter of a Regular Hexagon

A regular hexagon is a polygon with 6 equal sides and angles.

Perimeter of a regular hexagon = 6 × Length of one side

Example

Find the perimeter of a hexagon having side length 15cm.

Solution

Perimeter of a regular hexagon = 6 × Length of one side

Perimeter of a regular hexagon = 6 × 15

= 90 cm

Perimeter of a Regular Octagon

A regular octagon is a polygon with 8 equal sides and angles.

Perimeter of a regular octagon = 8 × length of one side

Example

Find the perimeter of an octagon having side length 7cm.

Solution

Perimeter of a regular octagon = 8 × length of one side

Perimeter of a regular octagon = 8 × 7

= 56 cm

Area

Area refers to the surface enclosed by a closed figure.

To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.

Here one square is taken as 1 sq.unit.

Example

Find the area of the given figure. (1 square = 1 m²)

Solution

The given figure is made up of line segments and is covered with some full squares and some half squares.

Full squares in figure = 32

Half squares in figure = 21

Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.

Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.

Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.

Area of a Rectangle

Area of a rectangle = (length × breadth)

Example

Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.

Solution

Length of the rectangle = 20 cm

Breadth of the rectangle = 12 cm

Area of the rectangle = length × breadth

= 20 cm × 12 cm

= 240 sq cm.

To find the length of a rectangle if breadth and area are given:

Example

What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?

Solution

Length = 48/6

 = 8 m

To find the breadth of the rectangle if length and area are given:

Example

What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?

Solution

Breadth = 81/8

= 9 m

Area of a Square

Area of a square is the region covered by the boundary of a square.

Area of a square = side × side

Example

Calculate the area of a square of side 13 cm.

Solution

Area of a square = side × side

= 13 × 13

= 169 cm².

Data

Data is a collection of raw facts and figures that give you information.

Recording Data

Recording of data depends upon the requirement of the data. Everybody has different ways to record data.

If we have to compare the choice of the people about certain movies then we have to collect the data of the survey which tells the choice of the people about those movies.

Organization of Data

Raw data is difficult to read, so we have to organize it in such a way so that we can use it in need.

Data can be organized in a tabular form.

Data is represented in a tabular form using frequency distribution and the tally marks.

Frequency tells the number of times the particular observation happened.

Tally marks are used to show the frequency of the data.

Tally marks are represented as

Example

There are 30 students in a class. They have to choose one sport each for the sports period.5 took badminton, 10 took cricket, 4 took football, 1 took hockey, 3 took tennis and 7 went for volleyball. Represent this data in the frequency distribution table.

Solution

To make a frequency distribution table-

• Make a table with three columns.
• Write the name of sports in the first column.
• Write the respective frequencies in front of each sport.
• Mark the tally marks according to the frequency given.

Pictograph

If we represent the data with the pictures of objects instead of numbers then it is called Pictograph. Pictures make it easy to understand the data and answer the questions related to it by just seeing it.

We can easily answer the questions like who has a maximum number of toys, who has the least number of toys etc.

Interpretation of a Pictograph

In the pictograph, we have to understand it and get the information from the pictures given.

If we have to represent more number of items then we can use the key which represents more numbers with one picture.

Example

The number of cars parked in a parking lot every day is given in the pictograph.

• Find the day when the highest number of cars are parked and how many?
• When the least number of cars did park?

Solution

In the above pictograph one car represent 5 cars.

• As there is the maximum number of cars is shown on Tuesday so the highest number of cars was parked on Tuesday.

Hence, 40 cars were parked on Tuesday in the parking lot.

• Least number of cars were parked on Monday as there are only 4 pictures of cars are shown on that day.

Drawing a Pictograph

Drawing a pictograph is an interesting task but it may be difficult to draw some difficult pictures repeatedly as we had used cars in the above example so we can use easy symbols to draw a pictograph.

We must use a proper key of the symbols so that it could be easily understood by anyone.

Example

The following table shows the choice of the fruits of the 35 students of class 3.represent the data in a pictograph.

Solution

Bar Graph

As the pictograph is a very time-consuming process, so we can use another way to represent data.

If we use the bars of the same width with equal spacing to represent the data in which the length of the bars represent the frequency is called Bar Graph or Bar Diagram.

Interpretation of a Bar Graph

Example

The following graph tells the favourite colours in a class of 30 students.

Answer the following questions:

• Which colour is liked by the maximum number of students?
• Which colour is liked by the same number of students?

Solution

• The graph shows that the pink colour is liked by 9 students so it is the favourite colour of the maximum number of students.
• Blue and green colour bars are equal in length and both are liked by 6 students.

Drawing a Bar Graph

Drawing a bar graph is an interesting task, but we must choose an appropriate scale to draw the bar graph. It depends upon our own choice that what we are taking for the scale.

Example

The daily sale of mobile phones in Vicky production is given below. Draw the bar graph to represent the data.

Solution

Steps to make a bar graph-

• Draw two lines, one horizontal and one vertical in L shape.
• Mark days on the horizontal line and no. of phones on the vertical line.
• Take a suitable scale for the number of phones and mark on the vertical line. Let 1 unit = 10 phones.
• Use the bars of equal width and draw them with the frequency given at the same distance.

The height of the bars tells the sale of the mobile phones in Vicky production.

This same bar graph can be made by interchanging the positions of the days and the number of phones.

When we use dots to write some numbers then that dot is the decimal point. This is used to show the part of a whole number.

Tenths

As we know that 1 cm = 10 mm, so if we have to find the opposite then

1mm = 1/10 cm or one-tenth cm or 0.1 cm.

Hence, the first number after the decimal represents the tenth part of the whole.

This reads as “thirty-four point seven”.

Representation of Decimals on Number Line

To represent decimals on the number line we have to divide the gap of each number into 10 equal parts as the decimal shows the tenth part of the number.

Example

Show 0.3, 0.5 and 0.8 on the number line.

Solution

All the three numbers are greater than 0 and less than 1.so we have to make a number line with 0 and 1 and divide the gap into 10 equal parts.

Then mark as shown below.

Fractions as Decimals

It is easy to write the fractions with 10 as the denominator in decimal form but if the denominator is not 10 then we have to find the equivalent fraction with denominator 10.

Example

Convert 12/5 and 3/2 in decimal form.

Solution

Decimals as Fractions

Example

Write 2.5 in a fraction.

Solution

Hundredths

As we know that 1 m = 100 cm, so if we have to find the opposite then

1 cm = 1/100 m or one-hundredth m or 0.01 m.

Hence, the second numbers after the decimal represent the hundredth part of the whole.

It reads as “thirteen point nine five”.

Decimal in the hundredth form shows that we have divided the number into hundred equal parts.

Example

If we say that 25 out of 100 squares are shaded then how will we write it in fraction and decimal form?

Solution

25 is a part of 100, so the fraction will be 25/100.

In the decimal form we will write it as 0.25.

Place Value Chart

This is the place value chart which tells the place value of each digit in the decimal number. It makes it easy to write numbers in decimal form.

Example

With the given place value chart write the number in the decimal form.

Solution

According to the above table-

Comparing Decimals

1. If the whole number is different.

If the whole numbers of the decimals are different then we can easily compare them .the number with the greater whole number will be greater than the other.

Example

Compare 532.48 and 682.26.

Solution:

As the whole numbers are different, so we can easily find that the number with a greater whole number is greater.

Hence 682.26 > 532.48

2. If the whole number is the same

If the whole numbers of the decimals are same then we will compare the tenth and then the hundredth part if required.

The number with the greater tenth number is greater than the other.

Example

Compare 42.36 and 42.68.

Solution

As the whole number is the same in both the numbers so we have to compare the tenth part.

Hence 42.68 > 42.36

Using Decimals

Generally, decimals are used in money, length and weight.

1. Money

Example: 1

Write 25 paise in decimals.

Solution:

100 paise = 1 Rs.

1 paise = 1/100 Rs. = 0.01 Rs.

25 paise = 25/100 Rs. = 0.25 Rs.

Example: 2

Write 7 rupees and 35 paise in decimals.

Solution:

7 rupees is the whole number, so

7 + 35/100 = 7 + 0.35 = 7.35 Rs.

2. Length

Example

If the height of Rani is 175 cm then what will be her height in meters?

Solution

100 cm = 1 m

1 cm = 1/100 m = 0.01 m

175 cm = 175/100 m

Hence, the height of Rani is 1.75 m.

3. Weight

Example

If the weight of a rice box is 4725 gram then what will be its weight in a kilogram?

Solution

1000 gm = 1 kg

1 gm = 1/1000 kg = 0.001 kg

Addition of Decimal numbers

To add the decimal numbers we can add them as whole numbers but the decimal will remain at the same place as it was in the given numbers. It means that we have to line up the decimal point in each number while writing them, and then add them as a whole number.

Example: 1

Add 22.3 and 34.1

Solution:

Write the numbers as given below, and then add them.

Example: 2

Add 1.234 and 4.1.

Solution:

There are three numbers after decimal in one number and one number after decimal in another number. So we should not get confused and write the numbers by lining up the decimal points of both the numbers, then add them.

Another way is to write the numbers in the place value chart, so that it will be easy to identify, how to write numbers.

Subtraction of Decimal Numbers

Subtraction is also done as normal whole numbers after lining up the decimals of the given number.

Example

Subtract 243.86 from 402.10.

Solution

• Write the numbers in a line so that the decimal points of both the numbers lined up.
• Then subtract and borrow as we do in whole numbers.
• Line up the decimal point in the answer also.

A Fraction is a part of whole. The ‘whole’ here could be an object or the group of objects. But all the parts of the whole must be equal.

1. The first one is the whole i.e. a complete circle.
2. In the second circle, if we divide the circle into two equal parts then the shaded portion is the half i.e. ½ of the circle.
3. In the third circle, if we divide the circle into four equal parts and shade only one part then the shaded part is the one fourth i.e. ¼ of the whole circle.
4. In the fourth circle, if we divide the circle into four equal parts and shade three parts then the shaded part is the three fourth i.e. ¾ of the whole circle.

Numerator and Denominator

The upper part of the fraction is called Numerator. It tells the number of parts we have.

The lower part of the fraction is called Denominator. It tells the total parts in a whole.

It reads as “three-fifths”.

Representation of fraction on Number line

Example

Solution

• Draw a number line.
• We know that ½ is less than 1 and greater than 0, so we have to divide the gap between two equal parts and then mark the middle point as ½.

As the denominator is the whole and the numerator is the part, so we have to divide the gap between 0 and 1 in the number of parts as the denominator is given.

For 1/3, divide into 3 equal parts.

For ¼, divide into 4 equal parts and so on.

Proper Fractions

If the numerator is less than the denominator then it is called proper fraction. If we represent a proper fraction on the number line than it will always lie between 0 and 1.

Examples

Improper fractions and Mixed fractions

When the numerator is greater than the denominator then it is called Improper fraction.

The above fraction is made by adding one whole part and one-fourth part.

The fraction made by the combination of whole and a part is called Mixed fraction.

Convert Mixed fraction into Improper fraction

A mixed fraction is in the form of 

We can convert it in the form of an improper fraction by

Example

Solution

Equivalent Fractions

Equivalent fractions are those fractions which represent the same part of a whole.

All the above images are different but equivalent fractions as they represent the same i.e. half part of a whole circle.

Finding equivalent fractions

1. Multiplying the same number

If we multiply the numerator and denominator of any fraction with the same number then we will get the equivalent fraction. There could be more than one equivalent fractions of one fraction.

Example

Find three equivalent fraction of ½.

Solution

1. Dividing the same number

If we divide the numerator and denominator of any fraction with the same number then we will get the equivalent fraction.

Example

Find the equivalent fraction of 18/27 with denominator 9.

Solution

To get the denominator 9 we need to divide it by 3.

So to find the equivalent fraction we need to divide the fraction by 3.

Hence the equivalent fraction with denominator 9 is 6/9.

The simplest form of a Fraction

If the numerator and denominator do not have any other common factor than 1 then it is said to be the simplest or lowest form of that fraction.

Example–

To find the equivalent fraction which is the simplest form we have to find the HCF of numerator and denominator and then divide them both by that HCF.

Example

Reduce the fraction 18/27 in the simplest form.

Solution

HCF of 18 and 27 is 9.

Hence,

2/3 is the lowest form of 18/27.

Like Fractions and Unlike Fractions

Fractions which have same denominators are known as Like fractions.

Example

Fractions which have different denominators are known as unlike fractions.

Example

Comparing fractions

If we have to compare the above two fractions then it is easy as the first one is less than 3 and the second one is greater than 3. So we can clearly say that

But sometimes it is not easy to compare it so easily. So we need some accurate procedure.

1. Comparing like fractions

Like fractions are the fractions with the same denominator so we have to compare them with the numerator only. The fraction with greater numerator is greater.

In the above example, both are divided into 8 equal parts, so the fraction with seven shaded part is greater than the 5 shaded parts.

1. Comparing unlike fractions

The fractions with different denominators are unlike fractions.

1. Unlike fraction with the same numerator

If we have to compare the fractions with different denominator but same numerator, we have to compare with the denominator only.

In that case, the fraction with the small denominator is greater than the other.

Example

Here the numerator is same i.e.3 so we will compare with the denominator.

The fraction with small denominator i.e. ¾ is greater than the fraction with the large denominator i.e. 3/8.

1. Unlike fraction with different numerators

If the numerator and denominator both are different then we have to make the denominator same by finding the equivalent fraction of both the fractions then compare the fractions as like fractions.

To find the equivalent fraction of both the fractions with the same denominator, we have to take the LCM of the denominator.

Example

Compare 6/7 and 3/5.

Solution

The product of 7 and 5 is 35.

So we will find the equivalent fraction of both the fractions with the denominator 35.

Now we can compare them as like fractions.

Addition and Subtraction of Fractions

1. Adding like fractions

In case of like fractions, the denominator is same so we can add them easily.

Steps to add like fractions-

1. Add the numerators.
2. Leave the common denominator same. (Don’t add the denominator).
3. Write the answer as

Example

Solution

1. Subtracting like fractions

Steps to subtract the like fractions-

1. Subtract the small numerator from the bigger one.
2. Leave the common denominator same.
3. Write the answer as

Example

Solution

1. Adding unlike fraction

If we have to add the unlike fractions, first we have to find the equivalent fraction of the given fractions with the same denominator then add them.

Steps to add unlike fractions-

1. Take the LCM of the denominator of the given fractions.
2. Find the equivalent fractions of both fractions with LCM as the denominator.
3. Add them as the like fractions.

Example

Solution

Take the LCM of 5 and 8, which is 40.

1. Subtracting unlike fractions

Steps to subtract unlike fractions-

1. Take the LCM of the denominator of the given fractions.
2. Find the equivalent fractions of both fractions with LCM as the denominator.
3. Subtract them as the like fractions.

Example

Solution

LCM of 4 and 5 is 20.

Remark: To add or subtract the mixed fractions, simply convert them in the improper fraction then add and subtract them directly

Th ere are so many situations where we have to use negative numbers. Negative Numbers are the numbers with the negative sign. These numbers are less than zero.

Example

We use negative numbers to represent temperature.

Where +10 shows 10° hotter than 0 and -10 shows 10° colder than 0.

Successor and Predecessor

If we move 1 to the right then it gives the successor of that number and if we move 1 to the left then it gives the predecessor of that number.

Tag me with a sign

• In the case of accounting we use negative sign to represent the loss and positive to represent the profit.
• In the case of sea level, we use a negative sign to represent the height of the place below the sea level and positive sign to represent the place above the sea level.

Integers

The collection of whole numbers and negative numbers together is called the Integers.

All the positive numbers are positive integers and all the negative numbers are negative integers. Zero is neither a positive nor a negative integer.

Representation of Integers on Number Line

To represent the integers on a number line, first, we have to draw a line and mark a point zero on it.

Then mark all the positive integers on the right side with the same distance as 1, 2, 3… and the entire negative numbers on the left side as -1,-2,-3…

Example

To mark (-7) we have to move 7 points to the left of zero.

Ordering of Integers

From the above number line, we can see that as we go to the right side the numbers are getting larger and as we move to the left the numbers are getting smaller.

Hence, any number on the right side on the number line is greater than the number on its left.

Example

5 is to the right of 2 so 5>2.

4 is to the right of -2, so 4>-2.

-4 is to the left of -1, so -4 < -1.

Some facts about Integers

• Any positive integer is always greater than any negative integer.
• Zero is less than every positive integer.
• Zero is greater than every negative integer.
• Zero doesn’t come in any of the negative and positive integers.

Addition of Integers

1. Addition of Two Positive Integers

If you have to add two positive integers then simply add them as natural numbers.

(+6) + (+7) = 6 + 7 = 13

2. Addition of Two Negative Integers

If we have to add two negative integers then simply add them as natural numbers and then put a negative sign on the answer.

(-6) + (-7) = – (6+7) = -13

3. Addition of One Negative and One Positive Integer

If we have to add one negative and one positive integer then simply subtract the numbers and put the sign of the bigger integer. We will decide the bigger integer ignoring the sign of the integers.

• (-6) + (7) = 1 (bigger integer 7 is positive integer)
• (6) + (-7) = -1(bigger integer 7 is negative integer)

Addition of Integers on a Number Line

1. Addition of Two Positive Integers

Example

Add 3 and 4 on the number line.

Solution

To add 3 and 4, first, we move 3 steps to the right of zero then again move 4 steps to the right from point 3.

As we reached to the point 7, hence (+3) + (+4) = +7

This shows that the sum of two positive integers is always positive.

2. Addition of Two Negative Integers

Example

Add (-2) and (-5) using a number line.

Solution

To add (-2) and (-5), first we move 2 steps to the left of zero then again move 5 steps to the left of (-2).

As we reached to the point (-7), hence (-2) + (-5) = -7.

This shows that the sum of two negative integers is always negative.

3. Addition of One Negative and One Positive Integer

a. If a positive integer is greater than the negative integer

To add (+6) and (-2), first we have to move 6 steps to the right from zero then move 2 steps to the left of point 6.

As we reached to the point 4, hence (+6) + (-2) = +4

b. If a negative integer is greater than the positive integer

To add (-5) and (+4), first we have to move 5 steps to the left of zero then move 4 steps to the right from point (-5).

As we reached to the point -1, hence (-5) + (+4) = (-1)

4. Additive Inverse

If we add numbers like (-7) and 7 then we get the result as zero. So these are called the Additive inverse of each other.

If we add (-2) + (2), then first we move 2 steps to the left of zero then we move two steps to the right of (-2).so finally we reached to zero.

Hence, if we add the positive and negative of the same number then we get the zero.

Example

What is the additive inverse of 4 and (-8)?

Solution

The additive inverse of 4 is (-4).

The additive inverse of (-8) is 8.

Subtraction of Integers on Number Line

If we subtract an integer from another integer then we simply add the additive inverse of that integer.

(-3) – (-2) = (-3) + 2 = -1

(-3) – (+2) = (-3) + (-2) = -5

1. Subtraction of Two Positive Integers

Example

Subtract 2 from 5.

Solution

To subtract 2 from 5, first, we move 5 steps to the right from zero then move 2 steps back to the left.

As we reached to 3 hence, 5 – (+2) = 5 – 2 = 3

2. Subtraction of Two Negative Integers

Example

Subtract (-12) from (-8).

Solution

To subtract (-12) from (-8), first, we have to move 8 steps to the left of zero then move 12 steps to the right of (-8).

As we reached to 4, hence (-8) – (-12) = (-8) + (12) = 4

3. Subtraction of One Negative and One Positive Integer

a. To subtract a positive integer from any other integer.

Example

Subtract 3 from (-4)

Solution

To subtract (-4) from (3), first, we have to move 4 steps to the left of zero then move 3 steps more to the left.

As we reached to (-7), hence (-4) – (+3) = (-4) + (-3) = -7.

b. To subtract a negative integer from any other integer

Example

Subtract (-3) from (4)

Solution

To subtract (-3) from (4), first, we have to move 4 steps to the right of zero then move 3 steps more to the right.

As we reached to (7), hence (4) – (-3) = (4) + (+3) = +7

The re are so many shapes around us made up of lines and curves like line segments, angles, triangles, polygons and circles etc. These shapes are of different sizes and measures.

Measuring Line Segments

A line segment is a fixed part of the line, so it must have some length. We can compare any line segment on the basis of their length.

1. Comparison by Observation

We can tell which line segment is greater than other just by observing the two line segments but it is not sure.

Here we can clearly say that AB > CD but sometimes it is difficult to tell which one is greater.

2. Comparison by Tracing

In this method we have to trace one line on paper then put the traced line segment on the other line to check which one is greater.

But this is a difficult method because every time to measure the different size of line segments we have to make a separate line segment.

3. Comparison using Ruler and a Divider

We can use a ruler to measure the length of a line segment.

Put the zero mark at point A and then move toward l to measure the length of the line segment, but it may have some errors on the basis of the thickness of the ruler.

This could be made accurate by using a Divider.

1. Put the one end of the divider on point A and open it to put another end on point B.
2. Now pick up the divider without disturbing the opening and place it on the ruler so that one end lies on “0”.
3. Read the marking on the other end and we can compare the two line.

Angles – “Right” and “Straight”

We can understand the concept of right and straight angles by directions.

There are four directions-North, South, East and West.

When we move from North to East then it forms an angle of 90° which is called Right Angle.

When we move from North to South then it forms an angle of 180° which is called Straight Angle.

When we move four right angles in the same direction then we reach to the same position again i.e. if we make a clockwise turn from North to reach to North again then it forms an angle of 360° which is called a Complete Angle.This is called one revolution.

In a clock, there are two hands i.e. minute hand and hour hand, which moves clockwise in every minute. When the clock hand moves from one position to another then turns through an angle.

When a hand starts from 12 and reaches to 12 again then it is said to be completed a revolution.

Acute, Obtuse and Reflex Angles

There are so many other types of angles which are not right or straight angles.

Angles	Meaning	Image
Acute Angle	An angle less than the right angle is called Acute angle.
Obtuse Angle	An angle greater than a right angle and less than straight angle is called Obtuse angle.
Reflex Angle	Angle greater than the straight angle is called Reflex angle.

Measuring Angles

By observing an angle we can only get the type of angle but to compare it properly we need to measure it.

An angle is measured in the “degree”. One complete revolution is divided into 360 equal parts so each part is one degree. We write it as 360° and read as “three hundred sixty degrees”.

We can measure the angle using a ready to use device called Protractor.

It has a curved edge which is divided into 180 equal parts. It starts from 0° to 180° from right to left and vice versa.

To measure an angle using protractor-

• Place the protractor on the angle in such a way that the midpoint of protractor comes on the vertex B of the angle.
• Adjust it so that line BC comes on the straight line of the protractor.
• Read the scale which starts from 0° coinciding with the line BC.
• The point where the line AB comes on the protractor is the degree measure of the angle.

Hence, ∠ABC = 72°

Perpendicular Lines

If two lines intersect with each other and form an angle of 90° then they must be perpendicular to each other.

Here AB and MN are intersecting at point N and form a right angle. We will write it as

AB ⊥ MN or MN ⊥ AB

Reads as AB is perpendicular to MN or MN is perpendicular to AB.

Perpendicular Bisector

If a perpendicular divides another line into two equal parts then it is said to be a perpendicular bisector of that line.

Here, CD is the perpendicular bisector of AB as it divides AB into two equal parts i.e. AD = DB.

Classification of Triangles

Triangle is a polygon with three sides. It is the polygon with the least number of sides. Every triangle is of different size and shape. We classify them on the basis of their sides and angles.

1. Classification on the basis of sides

Triangle	Meaning	Image
Scalene	If all the sides are different then it is called scalene triangle.
Isosceles	If two sides are equal then it is called isosceles triangle.
Equilateral	If all the sides are equal then it is called equilateral triangle.

2. Classification on the basis of Angles

Triangle	Meaning	Image
Acute Angled Triangle	If all the angles are less than 90° then this is called the acute-angled triangle.
Right Angled Triangle	If one of the angles is 90°then it is called the right-angled triangle.
Obtuse-angled Ariangle	If one of the angles of the triangle is obtuse angle then it is called Obtuse angled triangle.

Quadrilaterals

A polygon with four sides is called Quadrilateral.

S.No.	Name	Properties	Image
1.	Rectangle	It has two pairs of equal opposite sides.Opposite sides are parallel.All the angles are the right angle.
2.	Square	All the four sides are equal.Opposite sides are parallel.All the angles are the right angle.
3.	Parallelogram	It has two pairs of parallel opposite sides.Square and rectangle are also parallelograms.
4.	Rhombus	All the four sides are equal.Opposite sides are parallel.Opposite angles are equal.Diagonals intersect each other at the centre and at 90°.
5.	Trapezium	One pair of opposite sides is parallel.

Polygons

Any closed figure made up of three or more line segments is called Polygon.

We can classify the polygons on the basis of their sides and vertices –

Number of sides	Name of Polygon	Figure
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon
10	Decagon
n	n-gon

Three-dimensional Shapes

The solid shapes having three dimensions are called 3D shapes.

Some of the 3D shapes around us

Cone		ice-cream cone
Cube		Block
Cuboid		Match-box
Cylinder		Glass
Sphere		Ball
Pyramid		Rubrics in a pyramid shape

Faces, Edges and Vertices

All the flat surfaces of the solid shape are called the Faces of that figure.

The line segment where the two faces meet with each other is called Edge.

The point where the two edges meet with each other is called Vertex.

No. of Faces, Edges and Vertices in some common 3- D shapes

S.No.	3 – D shape	Figure	Faces	Edges	Vertices
1.	Cube		6	12	8
2.	Cuboid		6	12	8
3.	Cone		2	1	1
4.	Sphere		1	1	0
5.	Cylinder		3	2	0
6.	Square based Pyramid		6	8	5
7.	Triangular Prism		5	9	6
8.	Rectangular Prism		6	12	8