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
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 1st square = 4
No. of matchsticks used to make 2nd square = 7
No. of matchsticks used to make 3rd 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 50th 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.
No. of notebooks
1
2
3
4
…….
a
……..
Total cost
15
30
45
60
…….
15 a
…….
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
Statement
Expression
y subtracted from 12
12 – y
x multiplied by 6
6x
t Multiplied by 4, and then subtract 5 from the product.
4t – 5
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.
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.
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.
Name of fruit
Number of students
Mango
5
Apple
12
Guava
3
Litchi
7
Grapes
8
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.
Days
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
No. of phones sold
25
13
32
14
42
55
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.
Hundreds (100)
Tens (10)
Ones (1)
Tenths (1/10)
Hundredths (1/100)
4
6
3
8
5
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.
Ones (1)
Tenths (1/10)
Hundredths (1/100)
Thousandths (1/1000)
+
4
1
0
0
=
5
3
3
4
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.
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.
The first one is the whole i.e. a complete circle.
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.
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.
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
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
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.
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.
Comparing unlike fractions
The fractions with different denominators are unlike fractions.
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.
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
Adding like fractions
In case of like fractions, the denominator is same so we can add them easily.
Steps to add like fractions-
Add the numerators.
Leave the common denominator same. (Don’t add the denominator).
Write the answer as
Example
Solution
Subtracting like fractions
Steps to subtract the like fractions-
Subtract the small numerator from the bigger one.
Leave the common denominator same.
Write the answer as
Example
Solution
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-
Take the LCM of the denominator of the given fractions.
Find the equivalent fractions of both fractions with LCM as the denominator.
Add them as the like fractions.
Example
Solution
Take the LCM of 5 and 8, which is 40.
Subtracting unlike fractions
Steps to subtract unlike fractions-
Take the LCM of the denominator of the given fractions.
Find the equivalent fractions of both fractions with LCM as the denominator.
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.
Number
Predecessor
Successor
2
1
3
-8
-9
-7
-3
-4
-2
4
3
5
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.
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.
Put the one end of the divider on point A and open it to put another end on point B.
Now pick up the divider without disturbing the opening and place it on the ruler so that one end lies on “0”.
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
The term ‘Geometry’ is derived from the Greek word ‘Geometron’. This has 2 equivalents. ‘Geo’ means Earth and ‘metron’ means Measurement.
Points
It is a position or location on a plane surface, which are denoted by a single capital letter.
Line Segment
It is a part of a line with the finite length and 2 endpoints.
The points A and B are called the endpoints of the segment.
It is named as:
Line
It is made up of infinitely many points with infinite length and no endpoint.
It extends indefinitely in both directions.
Named as:
Or sometimes
Intersecting Lines
The two lines that share one common point are called Intersecting Lines.
This shared point is called the point of intersection.
Here, line l and m are intersecting at point C.
Real life example of intersecting lines:
Parallel Lines
Two or more lines that never intersect (Never cross each other) are called Parallel Lines.
Real life examples of parallel lines:
Ray
It is a part of a line with one starting point whereas extends endlessly in one direction.
Real life examples of the ray are:
Curves
Anything which is not straight is called a curve.
1. Simple Curve – A curve that does not cross itself.
2. Open Curve – Curve in which its endpoints do not meet.
3. Closed Curve – Curve that does not have an endpoint and is an enclosed figure.
A closed curve has 3 parts which are as follows
1. Interior of the curve
It refers to the inside/inner area of the curve.
The blue coloured area is the interior of the figure.
2. The exterior of the curve.
It refers to the outside / outer area of the curve.
The point marked A depicts the exterior of the curve.
3. The boundary of the curve
It refers to the dividing line thus it divides the interior and exterior of the curve.
The black line which is dividing the interior and exterior of the curve is the boundary.
The interior and boundary of the curve together are called the curves “region”.
Polygons
It is a 2d closed shape made of line segments / straight lines only.
Sides –It refers to the line segments which form the polygon, as in the above figure AB, BC, CD, DA are its sides.
Vertex – Point where 2 line segments meet, as in the above figure A, B, C and D are its vertices.
Adjacent Sides – If any 2 sides share a common endpoint they are said to be adjacent to each other thus called adjacent sides, as in the above figure AB and BC, BC and CD, CD and DA, DA and AB are adjacent sides.
Adjacent Vertices – It refers to the endpoints of the same side of the polygon. As in the above figure A and B, B and C, C and D, D and A are adjacent vertices.
Diagonals – It refers to the joins of the vertices which are not adjacent to each other. As in the above figure, AC and BD are diagonals of the polygon.
Angles
A figure formed from 2 rays which share a common endpoint is called Angle.
The rays forming the angle are known as its arms or sides.
The common endpoint is known as its vertex.
An angle is also associated with 3 parts
1. Interior – It refers to the inside/inner area.
The green coloured area is the interior of the angle.
2. Angle/boundary – It refers to the arms of the angle.
The red point is on the arm of the angle.
3. Exterior – It refers to the outside / outer area.
The blue point depicts the exterior of the figure.
Naming an Angle
While naming an angle the letter depicting the vertex appears in the middle.
Example
The above angle can also be named as ∠CBA.
An angle can also be named just by its vertex.
Example
Triangle
It is a 3 sided polygon. It is also the polygon with the least number of the sides.
Vertices: A, B and C
Sides: AB, BC and CA
Angles: ∠A, ∠B and ∠C
Here, the light blue coloured area is the interior of the angle.
The black line is the boundary.
Whereas, the dark blue area is the exterior of the angle.
Quadrilaterals
It is a 4 sided polygon
Vertices: A, B, C, D
Sides: AB, BC, CD, DA
Angle: ∠A, ∠B, ∠C, ∠D
Opposite Sides: AB and DC, BC and AD
Opposite Angles: ∠B and ∠D, ∠A and ∠C
Adjacent Angles: ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A.
Circles
It is a simple closed curve and is not considered as a polygon.
Parts of Circles
1. Radius – It is a straight line connecting the centre of the circle to the boundary of the same. Radii is the plural of ‘radius’.
2. Diameter –It is a straight line from one side of the circle to the other side passing through the centre.
3. Circumference – It refers to the boundary of the circle.
4. Chord – Any line that connects two points on the boundary of the circle is called Chord. Diameter is the longest chord.
5. Arc – It is the portion of the boundary of the circle.
6. Interior of the Circle – Area inside the boundary of the circle is called the Interior of the Circle.
7. The Exterior of the Circle – Area outside the boundary of the circle is called the Exterior of the Circle.
8. Sector– It is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.
9. Segment – It is the region in the interior of the circle enclosed by an arc and a chord.
Semi-circle
A diameter divides the circle into two semi-circles. Hence the semicircle is the half of the circle, which has the diameter as the part of the boundary of the semicircle.
The numbers which exactly divides the given number are called the Factors of that number.
As we can see that we get the number 12 by
1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2 and 12 ×1
Hence,
1, 2, 3, 4, 6 and 12 are the factors of 12.
The factors are always less than or equal to the given number.
Multiples
If we say that 4 and 5 are the factors of 20 then 20 is the multiple of 4 and 5 both.
List the multiples of 3
Multiples are always more than or equal to the given number.
Some facts about Factors and Multiples
1 is the only number which is the factor of every number.
Every number is the factor of itself.
All the factors of any number are the exact divisor of that number.
All the factors are less than or equal to the given number.
There are limited numbers of factors of any given number.
All the multiples of any number are greater than or equal to the given number.
There are unlimited multiples of any given numbers.
Every number is a multiple of itself.
Perfect Number
If the sum of all the factors of any number is equal to the double of that number then that number is called a Perfect Number.
Perfect Number
Factors
Sum of all the factors
6
1, 2, 3, 6
12
28
1, 2, 4, 7, 14, 28
56
496
1, 2, 4, 8, 16, 31, 62, 124, 248, 496
992
Prime Numbers
The numbers whose only factors are 1 and the number itself are called the Prime Numbers.
Like 2, 3, 5, 7, 11 etc.
Composite Numbers
All the numbers with more than 2 factors are called composite numbers or you can say that the numbers which are not prime numbers are called Composite Numbers.
Like 4, 6, 8, 10, 12 etc.
Remark: 1 is neither a prime nor a composite number.
Sieve of Eratosthenes Method
This is the method to find all the prime numbers from 1 to 100.
Step 1: First of all cross 1, as it is neither prime nor composite.
Step 2: Now mark 2 and cross all the multiples of 2 except 2.
Step 3: Mark 3 and cross all the multiples of 3 except 3.
Step 4: 4 is already crossed so mark 5 and cross all the multiples of 5 except 5.
Step 5: Continue this process until all the numbers are marked square or crossed.
This shows that all the covered numbers are prime numbers and all the crossed numbers are composite numbers except 1.
Even and Odd Numbers
All the multiples of 2 are even numbers. To check whether the number is even or not, we can check the number at one’s place. If the number at ones place is 0,2,4,6 and 8 then the number is even number.
The numbers which are not even are called Odd Numbers.
Remark: 2 is the smallest even prime number. All the prime numbers except 2 are odd numbers.
Tests for Divisibility of Numbers
1. Divisibility by 2:
If there are any of the even numbers i.e. 0, 2, 4, 6 and 8 at the end of the digit then it is divisible by 2.
Example
Check whether the numbers 63 and 240 are divisible by 2 or not.
Solution:
1. The last digit of 63 is 3 i.e. odd number so 63 is not divisible by 2.
2. The last digit of 240 is 0 i.e. even number so 240 is divisible by 2.
2. Divisibility by 3:
A given number will only be divisible by 3 if the total of all the digits of that number is multiple of 3.
Example
Check whether the numbers 623 and 2400 are divisible by 3 or not.
Solution:
1. The sum of the digits of 623 i.e. 6 + 2 + 3 = 11, which is not the multiple of 3 so 623 is not divisible by 3.
2. The sum of the digits of 2400 i.e. 2 + 4 + 0 + 0 = 6, which is the multiple of 3 so 2400 is divisible by 3.
3. Divisibility by 4:
We have to check whether the last two digits of the given number are divisible by 4 or not. If it is divisible by 4 then the whole number will be divisible by 4.
Example
Check the number 23436 and 2582 are divisible by 4 or not.
Solution:
1. The last two digits of 23436 are 36 which are divisible by 4, so 23436 are divisible by 4.
2. The last two digits of 2582 are 82 which are not divisible by 4 so 2582 is not divisible by 4.
4. Divisibility by 5:
Any given number will be divisible by 5 if the last digit of that number is ‘0′ or ‘5′.
Example
Check whether the numbers 2348 and 6300 are divisible by 5 or not.
Solution:
1. The last digit of 2348 is 8 so it is not divisible by 5.
2. The last digit of 6300 is 0 so it is divisible by 5.
5. Divisibility by 6:
Any given number will be divisible by 6 if it is divisible by 2 and 3 both. So we should do the divisibility test of 2 and 3 with the number and if it is divisible by both then it is divisible by 6 also.
Example
Check the number 342341 and 63000 are divisible by 6 or not.
Solution:
1. 342341 is not divisible by 2 as the digit at ones place is odd and is also not divisible by 3 as the sum of its digits i.e. 3 + 4 + 2 + 3 + 4 + 1 = 17 is also not divisible by 3.Hence 342341 is not divisible by 6.
2. 63000 is divisible by 2 as the digit at ones place is even and is also divisible by 3 as the sum of its digits i.e. 6 + 3 + 0 + 0 + 0 = 9 is divisible by 3.Hence 63000 is divisible by 6.
6. Divisibility by 7:
Any given number will be divisible by 7 if we double the last digit of the number and then subtract the result from the rest of the digits and check whether the remainder is divisible by 7 or not. If there is a large number of digits then we have to repeat the process until we get the number which could be checked for the divisibility of 7.
Example
Check the number 2030 is divisible by 7 or not.
Solution:
Given number is 2030
1. Double the last digit, 0 × 2 = 0
2. Subtract 0 from the remaining number 203 i.e. 203 – 0 = 203
3. Double the last digit, 3 × 2 = 6
4. Subtract 6 from the remaining number 20 i.e. 20 – 6 = 14
5. The remainder 14 is divisible by 7 hence the number 203 is divisible by 7.
7. Divisibility by 8:
We have to check whether the last three digits of the given number are divisible by 8 or not. If it is divisible by 8 then the whole number will be divisible by 8.
Example
Check whether the number 74640 is divisible by 8 or not.
Solution:
The last three digit of the number 74640 is 640.
As the number 640 is divisible by 8 hence the number 74640 is also divisible by 8.
8. Divisibility by 9:
Any given number will be divisible by 9 if the total of all the digits of that number is divisible by 9.
Example
Check whether the number 2320 and 6390 are divisible by 9 or not.
Solution:
1. The sum of the digits of 2320 is 2 + 3 + 2 + 0 = 7 which is not divisible by 9 so 2320 is not divisible by 9.
2. The sum of the digits of 6390 is 6 + 3 + 9 + 0 = 18 which is divisible by 9 so 6390 is divisible by 9.
9. Divisibility by 10:
Any given number will be divisible by 10 if the last digit of that number is zero.
Example
Check the number 123 and 2630 are divisible by 10 or not.
Solution:
1. The ones place digit is 3 in 123 so it is not divisible by 10.
2. The ones place digit is 0 in 2630 so it is divisible by 10.
Common Factors and Common Multiples
Example: 1
What are the common factors of 25 and 55?
Solution:
Factors of 25 are 1, 5.
Factors of 55 are 1, 5, 11.
Common factors of 25 and 55 are 1 and 5.
Example: 2
Find the common multiples of 3 and 4.
Solution:
Common multiples of 3 and 4 are 0, 12, 24 and so on.
Co-prime Numbers
If 1 is the only common factor between two numbers then they are said to be Co-prime Numbers.
Example
Check whether 7 and 15 are co-prime numbers or not.
Solution:
Factors of 7 are 1 and 7.
Factors of 15 are 1, 3, 5 and 15.
The common factor of 7 and 15 is 1 only. Hence they are the co-prime numbers.
Some more Divisibility Rules
1. Let a and b are two given numbers. If a is divisible by b then it will be divisible by all the factors of b also.
If 24 is divisible by 12 then 24 will be divisible by all the factors of 12(i.e.2, 3, 4, 6) also.
2. Let a and b are two co-prime numbers. If c is divisible by a and b then c will be divisible by the product of a and b (ab) also.
If 24 is divisible by 2 and 3 which are the co-prime numbers then 24 will also be divisible by the product of 2 and 3 (2×3=6).
3. If a and b are divisible by c then a + b will also be divisible by c.
If 24 and 12 are divisible by 4 then 24 + 12 = 36 will also be divisible by 4.
4. If a and b are divisible by c then a-b will also be divisible by c.
If 24 and 12 are divisible by 4 then 24 -12 = 12 will also be divisible by 4.
Prime Factorisation
Prime Factorisation is the process of finding all the prime factors of a number.
There are two methods to find the prime factors of a number-
1. Prime factorisation using a factor tree
We can find the prime factors of 70 in two ways.
The prime factors of 70 are 2, 5 and 7 in both the cases.
2. Repeated Division Method
Find the prime factorisation of 64 and 80.
The prime factorisation of 64 is 2 × 2 × 2 × 2 × 2 × 2.
The prime factorisation of 80 is 2 × 2 × 2 × 2 × 5.
Highest Common Factor (HCF)
The highest common factor (HCF) of two or more given numbers is the greatest of their common factors.
Its other name is (GCD) Greatest Common Divisor.
Method to find HCF
To find the HCF of given numbers, we have to find the prime factorisation of each number and then find the HCF.
Example
Find the HCF of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
HCF of 60 and 72 is 2 × 2 × 3 = 12.
Lowest Common Multiple (LCM)
The lowest common multiple of two or more given number is the smallest of their common multiples.
Methods to find LCM
1. Prime Factorisation Method
To find the LCM we have to find the prime factorisation of all the given numbers and then multiply all the prime factors which have occurred a maximum number of times.
Example
Find the LCM of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
To find the LCM, we will count the common factors one time and multiply them with the other remaining factors.
LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 = 360
2. Repeated Division Method
If we have to find the LCM of so many numbers then we use this method.
Example
Find the LCM of 105, 216 and 314.
Solution:
Use the repeated division method on all the numbers together and divide until we get 1 in the last row.
LCM of 105,216 and 314 is 2 × 2 × 2 × 3 × 3 × 3 × 5 × 7 × 157 = 1186920
Real life problems related to HCF and LCM
Example: 1
There are two containers having 240 litres and 1024 litres of petrol respectively. Calculate the maximum capacity of a container which can measure the petrol of both the containers when used an exact number of times.
Solution:
As we have to find the capacity of the container which is the exact divisor of the capacities of both the containers, i. e. maximum capacity, so we need to calculate the HCF.
The common factors of 240 and 1024 are 2 × 2 × 2 × 2. Thus, the HCF of 240 and 1024 is 16. Therefore, the maximum capacity of the required container is 16 litres.
Example: 2
What could be the least number which when we divide by 20, 25 and 30 leaves a remainder of 6 in every case?
Solution:
As we have to find the least number so we will calculate the LCM first.
LCM of 20, 25 and 30 is 2 × 2 × 3 × 5 × 5 = 300.
Here 300 is the least number which when divided by 20, 25 and 30 then they will leave remainder 0 in each case. But we have to find the least number which leaves remainder 6 in all cases. Hence, the required number is 6 more than 300.
