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.
The method in which we first find the value of a unit quantity and then use it to find the value of any required quantity is called the unitary method. The unitary method can be used to solve problems related to distance, time speed, and calculating the cost of materials. The unitary method is used for various applications.
The unitary method consists of two type of variations:
Two quantities are said to be in direct variation if one quantity increases, then the other also increases or when one quantity decreases, the other also decreases.
Two quantities are said to be inverse variation if,
On increasing one quantity, the other quantity decreases.
On decreasing one quantity, the other quantity increases.
Let us consider some examples:
Example 1:The cost of 15 pens is Rs 360, What is the cost of 8 such pens?
Solution:
Cost of 15 pens = Rs, 360.
Cost of 1 pen = Rs. 360/15.
Cost of 8 pen = (360/15) * 8 = Rs 192.
Example 2:18 men can make 90 identical tables in one day. Find how many men will make 20 such tables in one day?
Solution:
In one day, 90 tables are made by 18 men.
In one day, 1 tables are made by 18/1 men.
In one day, 20 tables are made by (18/1) * 20 men.
Example 3: A car running with uniform speed covers a distance of 96 km in 3 hours. How much distance will the car cover in 5 hours running with the same speed?
Solution:
In 3 hours, car covers 96 km.
In 1 hours, car covers km = (96/3) = 32 km.
In 5 hours, car covers = 32 * 5 = 170 km.
Example 4: A car can travel 360 km consuming 24 litres of petrol. How much petrol will it consume while travelling through a distance of 480 km?
Solution:
The car can travel 360 km consuming 24 litres of petrol.
The car can travel 1 km consuming (24/360)km.
The car can travel 480 km consuming = (24/360) * 480 = 32 litres.
Finding and understanding patterns gives us great power. With patterns we can learn to predict the future, discover new things and better understand the world around us.
And playing with patterns is fun.
Arithmetic Sequences
An Arithmetic Sequence is made by adding the same value each time.
Example
1, 4, 7, 10, 13, 16, 19, 22, 25, …
This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time, like this:
Geometric Sequences
A Geometric Sequence is made by multiplying by the same value each time.
1, 3, 9, 27, 81, 243, …
This sequence has a factor of 3 between each number. The pattern is continued by multiplying by 3 each time, like this:
The Fibonacci Sequence is found by adding the two numbers before it together. The 2 is found by adding the two numbers before it (1+1) The 21 is found by adding the two numbers before it (8+13) The next number in the sequence above would be 55 (21+34)
A number line is a picture of a graduated straight line. The integers are shown as marked points evenly spaced on the number line. The line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is used to help in teaching simple addition and subtraction, which involves negative numbers. A number line is usually represented as being horizontal. According to one custom, positive numbers always lie on the right side of zero, negative numbers always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions.
Now, let us understand some terms.
Integers: The collection of the numbers, that is, ? -3, -2, -1, 0, 1, 2, 3, ?., is called integers.
Absolute value: The distance of a rational number from zero on the number line is called its absolute value.
Natural numbers: Numbers by which we can count things in nature are called natural numbers.
Whole numbers: Natural number along with zero forms a collection of whole numbers.
Let us understand number line:
If you move towards the right from the zero mark on the number line, the value of thenumbers increases. If you move towards the left from the zero mark on the number line, the value of the numbers decreases.
A number line starting from 1 and marked 2, 3, 4, 5 at equal distances on the right side of 1 is called number line representing natural numbers.
A number line starting from 0 and marked 1, 2, 3, 4, 5 at equal distances on the right side of 0 is called number line representing whole numbers.
A number line with 0 marked anywhere on it with positive numbers 2, 3, 4, 5 marked on the right side of 0 at equal distances and negative numbers -1, -2, 4, 5 marked on the left side of ) is number line representing integers.
Let us consider some examples:
Example 1: Which is greater?
a) -12 and 15
b) -15 and -23
Solution:
a) 15
b) -15
Example 2: Replace * with the signs > or < in the given statements.
Place value is one of the fundamental concepts in mathematics.It is important as it helps students to understand the meaning of a number. Place value is needed to understand the order of numbers as well. The concept that numbers can be broken apart and put back together gives the student a better understanding of how different mathematical operations work. It will be easy for the student to carry out operations such as addition, subtraction, multiplication, division, expanded notation, etc.
Place value of any digit is the value of digit according to its position in the number.
Place value of a digit depends upon the position it occupies in the number.
Largest number of n digit + 1 = smallest number of (n + 1) digits.
Smallest number of n digit – 1 = Largest number of (n – 1) digits.
A concrete number is a number which refers to a particular unit and is meaningful such as 8 meters, 12 kg etc.,
An abstract number is a number which does not refer to any particular unit such as 8, 12 etc.,
Let us consider some examples:
Example 1:
Write the place value of both the six in the number 36268 and find the sum of these values.
Solution:
In 36268, place value is 6000.
The other 6 at ten’s place, so its place value is 60.
Sum = 6000 + 60 = 6060
Example 2:
Write the place value of both the five in number 9,45,582 and find the difference of these place values.
Solution:
In 9,45,582, place value is 5000. The other 5 at hundred’s place, so its place value is 500.
Required sum = 5000 – 500 = 4500.
Example 3:
Find the place value of 7 in number 5731?
Solution:
Place value of 7 is 700.
Example 4:
Write the largest 4 digit number having 3 in tens place?
Before the invention of numbers, counting was done using some sort of physical objects such as pebbles or sticks. The numbers came into existence, eventually and then the need for adapting to a standard system of counting.
The Hindu–Arabic numeral system also known as the Arabic numeral system or Hindu numeral system, is a positional decimal numeral system. It is the most common system for the symbolic representation of numbers in the world. It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. The system later spread to medieval Europe.
The system is based on ten different symbols. The symbols in actual use are descended from Brahmi numerals and have split into various typographical variants.
Today, this numerical system is still used worldwide.
Hindu Arabic system of numeration:
In Indian number system, ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used to write numeral. Each of this number is called a digit.
Values of the places in the Indian system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousand, Lakhs, Ten Lakhs, Crores and so on. The following place value chart can be used to identify the digit in any place in the Indian system.
Commas are placed to the numbers to help us read and write large numbers easily. As per Indian system of numeration, the first comma is placed after the hundreds place. Commas are then placed after every two digits.
Periods
Crores
lakhs
Thousands
Ones
Places
Tens
Ones
Tens
Ones
Tens
Ones
Hundreds
Tens
Ones
Example 1:
Using Hindu-Arabic system, read the number 850746
Solution:
850746 – Eighty Lakh Fifty Thousand Seven Hundred Forty Six. Place value chart is as shown in the picture below on the left:
Example 2:
Write four crores fifteen lakh fifty thousand five hundred twenty seven in the numeral form using the Hindu-Arabic system.
Solution:
Number – 4,15,50,527
The international system of numeration:
A numeral system (or system of numeration) is a system for expressing numbers; using digits or other symbols in a consistent manner. The number the numeral represents is called its value. The most commonly used system of numerals is the Hindu–Arabic numeral system which was invented by Indian mathematicians.
The International number system is another method of representing numbers. In the International numbering system also, different periods are formed to read the large numbers easily. The periods used here are ones, thousand and millions, etc.
The international system of numeration:
As per the International numeration system, the first comma is placed after the hundreds place. Commas are then placed after every three digits.
The values of the places in the International system of numeration are Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten millions and so on. 1 million = 1000 thousand, 1 billion = 1000 millions.
Shown on the left side is a chart with the International number system. Shown on the right side is a chart with a comparison between Indian and International system:
Let us consider some examples.
International system – table
Indian system – table
Let us consider some examples:
Example 1:Using the International system, write the number; Six million, four hundred and eleven thousand, two hundred and sixty.
Solution: 6,411,260.
Example 2: Using the international system, read the number 7456123.
Solution: 7456123- Seven million Four Hundred Fifty six thousand one hundred twenty three.
Example 3: Write the number in words: 12,367,169.
Solution: Twelve million, three hundred and sixty-seven thousand, one hundred and sixty-nine.
Example 4: Write seven hundred forty three million eight hundred thirteen thousand two hundred fifty six in the numeral form using the international system.
Solution: 743,813,256.
Similarly, 48670002 can be read as
48,670,002 – Forty eight million, six hundred and seventy thousand and two.
Data: A collection of numbers gathered to give some information.
Recording Data: Data can be collected from different sources.
Pictograph: The representation of data through pictures of objects. It helps answer the questions on the data at a glance.
Bar Graph: Pictorial representation of numerical data in the form of bars (rectangles) of equal width and varying heights.
We have seen that data is a collection of numbers gathered to give some information.
To get a particular information from the given data quickly, the data can be arranged in a tabular form using tally marks.
We learnt how a pictograph represents data in the form of pictures, objects or parts of objects. We have also seen how to interpret a pictograph and answer the related questions.
We have drawn pictographs using symbols to represent a certain number of items or
things.
We have discussed how to represent data by using a bar diagram or a bar graph. In a
bar graph, bars of uniform width are drawn horizontally or vertically with equal
spacing between them. The length of each bar gives the required information.
To do this we also discussed the process of choosing a scale for the graph.
For example, 1 unit = 100 students. We have also practiced reading a given bar graph. We have seen how interpretations from the same can be made.
We have discussed multiples, divisors, factors and have seen how to identify factors and multiples.
We have discussed and discovered the following:
(a) A factor of a number is an exact divisor of that number.
(b) Every number is a factor of itself. 1 is a factor of every number.
(c) Every factor of a number is less than or equal to the given number.
(d) Every number is a multiple of each of its factors.
(e) Every multiple of a given number is greater than or equal to that number.
(f) Every number is a multiple of itself.
We have learnt that –
(a) The number other than 1, with only factors namely 1 and the number itself, is a prime number. Numbers that have more than two factors are called composite numbers. Number 1 is neither prime nor composite.
(b) The number 2 is the smallest prime number and is even. Every prime number other than 2 is odd.
(c) Two numbers with only 1 as a common factor are called co-prime numbers.
(d) If a number is divisible by another number then it is divisible by each of the factors of that number.
(e) A number divisible by two co-prime numbers is divisible by their product also. We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,8,9 and 11. We have explored the relationship between digits of the numbers and their divisibility by different numbers.
(a) Divisibility by 2,5 and 10 can be seen by just the last digit.
(b) Divisibility by 3 and 9 is checked by finding the sum of all digits.
(c) Divisibility by 4 and 8 is checked by the last 2 and 3 digits respectively.
(d) Divisibility of 11 is checked by comparing the sum of digits at odd and even places. We have discovered that if two numbers are divisible by a number then their sum and difference are also divisible by that number.
We have learnt that –
(a) The Highest Common Factor (HCF) of two or more given numbers is the highest of their common factors.
(b) The Lowest Common Multiple (LCM) of two or more given numbers is the lowest of their common multiples.
