Positive Rational Number

In this article, you will learn about Positive Rational Numbers. Get to know about solved examples and explaining all about how come they are called Positive Rational Numbers. A Rational Number is said to be positive if both the numerator and denominator are either positive integers or negative integers. You can also say that a rational number is positive if both numerator and denominator are of the same sign.

1/6, 2/7, -9/-11, -5/-13, 8/12 are positive rationals, but 6/-5, -3/11, -8/7, 9/-23 are not positive rationals.

Is every natural number a positive rational number?

We know 1 = 1/1, 2 = 2/1, 3 = 3/1, 4 = 4/1 and ……..

Any natural number n can be written as n/1 where n, 1 are positive integers.

Therefore, every natural number is a positive rational number. Do remember Rational Number 0 is neither positive nor negative.

Determine whether the following rational numbers are positive or not

(i) -7/3

-7/ 3 is not a positive rational number as both the numerator and denominator are of opposite sign.

(ii) -9/-11

-9/-11 is a positive rational number as both the numerator and denominator are of the same sign.

(iii) 11/19

11/19 is a positive rational number since both numerator and denominator are positive integers.

(iv) 21/-7

21/-7 is not a positive rational number as both the numerator and denominator are of opposite sign.

(v) -105/7

-105/7 is not a positive rational number as both numerator and denominator are of opposite sign.

(vi) 25/31

25/31 is a positive rational number since both numerator and denominator are positive integers.

(vii) -6/5

-6/5 is not a positive rational number as both the numerator and denominator are of opposite sign.

(viii) 21/-25

21/-25 is not a positive rational number as both numerator and denominator are of opposite sign.

Thus, we can say that a rational number is positive if it has both numerator and denominator are of the same sign.

Equality of Rational Numbers using Cross Multiplication

Let us learn about Equality of Rational Numbers using Cross Multiplication in detail here. Check out the Procedure to determine whether Rational Numbers are Equal or not using the Cross Multiplication Technique. Have a glance at the solved examples explaining the concept in detail so that you can solve related problems.

How to determine Equality of Rational Numbers using Cross Multiplication?

There are numerous methods to check the equality of rational numbers. But here we are using the Cross Multiplication Method to check whether the given rational numbers are equal or not. Follow the guidelines to check the equality of rational numbers.

Let us consider two rational numbers a/b, c/d

a/b = c/d

⇔ a × d = b × c

⇔ The Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second

Solved Examples

1.  Determine whether the following pair of Rational Numbers are Equal or Not?

8/4 and 6/3

Given Rational Numbers are 8/4 and 6/3

⇔ We know a × d = b × c

Multiplying Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second we get

8*3 = 6*4

24 = 24

Therefore, the given rational numbers 8/4 and 6/3 are equal.

2. If -8/6 = k/30, find the value of k?

Solution:

-8/6 = k/30

Cross multiplying we get

-8*30 = k*6

Performing basic math we get the value of k

(-8*30)/6 =k

k=-40

Therefore, the value of k is -40.

3. If 5/m = 40/16 determine the value of m?

Solution:

5/m = 40/16

Cross multiplying we get 5*16 = m*40

Separating m to get the value of it.

m= (5*16)/40

= 80/40

= 2

Therefore, the value of m is 2.

4. Fill in the Blank -7/10 = …/120?

Solution:

In order to express -7 as a denominator with 120, we first need to find out the number which when multiplied by 10 gives 120.

Thus, the integer is 120÷ 10 = 12

Multiplying the numerator and denominator of a given rational number with 12 we get

-7/10 = (-7*12)/(10*12)

= -84/120

Thus, the required number is -84/120.

Equality of Rational Numbers with Common Denominator

You will learn about the Equality of Rational Numbers with Common Denominator from this article. Check out how to determine whether given Rational Numbers are equal or not using the Common Denominator. Apart from the procedure, we have even listed a few examples for better understanding.

How to determine Equality of Rational Numbers with Common Denominator?

There are many methods to check the equality of rational numbers but here in this article, we are going to discuss the method of the same denominator. We have listed the procedure on how to make denominators equal for the given rational numbers and they are in the following fashion.

Step 1: Check the given Rational Numbers.

Step 2: Multiply the numerator and denominator of the first number with the denominator of the second number.

Step 3: On the other hand, multiply the numerator and denominator of the second number with the denominator of the first number.

Step 4: Later check the numerators of the numbers obtained in steps 2, 3. If the numerators are equal then the given rational numbers are equal or else they are not equal.

Solved Examples

1. Are the Rational Numbers -7/5 and -5/3 equal?

Solution:

Given Rational Numbers are -7/5, -5/3

Multiply the second number numerator and denominator with the denominator of the first number i.e.

-5*5/3*5

= -25/15

Multiply the first number numerator and denominator with the denominator of the second number i.e.

-7*3/5*3

= -21/15

Check the numerators obtained in the earlier steps and see whether they are equal or not.

Since 21, 25 aren’t equal both the given rational numbers aren’t equal.

Therefore, -7/5 and -5/3 are not equal.

2. Show that the Rational Numbers -6/8 and -9/12  are equal?

Solution:

Given Rational Numbers are -6/8, -9/12

Multiply both numerator and denominator of the second number with the denominator of the first number.

= (-9/12)*8

= -72/96

Multiply both numerator and denominator of the first number with the denominator of the second number.

= (-6/8)*12

= -72/96

Therefore, Rational Numbers -6/8 and -9/12 are equal.

Equality of Rational Numbers using Standard Form

In this article of ours, we tried covering everything about the concept of Equality of Rational Numbers using Standard Form. You will find all about how to determine whether two rational numbers are equal or not. However, there are various methods to know whether given rational numbers are equal or not but we will employ the standard form method here.

How to determine the Equality of Rational Numbers using Standard Form?

To check whether the given rational numbers are equal or not you need to find out the standard form of both of them individually. If the standard form of both the rational numbers is equal then the rational numbers are equal or else not equal.

Solved Examples

1. Determine whether the Rational Numbers 4/-9 and -16/36 equal or not using a standard form?

Solution:

Given Rational Numbers are 4/-9, -16/36

Check for the denominators in both the rational numbers if they aren’t positive change them to positive.

4/-9 = 4*(-1)/-9*(-1)

=-4/9

GCD(4,9) = 1, thus -4/9 is the standard form.

-16/36 since it has a positive denominator it remains unchanged

Find the GCD of absolute values of the numerator and denominator for the rational numbers.

GCD(16, 36) = 4

To reduce the rational number to standard form divide both numerator and denominator with GCD obtained.

-16/36 = (-16÷4)/(36÷4)

= -4/9

-4/9 is the standard form of -16/36

Since the standard forms of rational numbers are equal both the given rational numbers 4/-9 and -16/36 are equal.

2. Determine whether the Rational Numbers 2/3 and 5/7 equal or not using a Standard Form?

Solution:

Given Rational Numbers are 2/3 and 5/7

Since both the denominators are positive you need not multiply or divide to make them positive.

Find the GCD of absolute values of numerator and denominator in the given rational numbers.

GCD(2, 3) =1

GCD(5, 7) = 1

Since both the rational numbers have GCD 1 and the numbers are relatively prime. Both the Rational Numbers are in Standard Form.

2/3 is not equal to 5/7

Therefore, Rational Numbers 2/3 and 5/7 are not equal.

Standard Form of a Rational Number

Standard Form of a Rational Number: A rational number is said to be in standard form if the common factor between numerator and denominator is only 1 and the denominator is always positive. Furthermore, the numerator can have a positive sign. Such Numbers are called Rational Numbers in Standard Form. Check out a few examples that illustrate the procedure of expressing Rational Number in Standard Form to be familiar with the concept even better.

What is the Standard Form of a Rational Number?

Usually, a rational number a/b is said to be in standard form if it has no common factors other than 1 between the numerator and denominator alongside the denominator b should be positive.

How to Convert a Rational Number into Standard Form?

Go through the below-listed guidelines to express a Rational Number into Standard Form. The Detailed Procedure is explained for better understanding and they are along the lines

Step 1: Have a look at the given rational number.

Step 2: Firstly, find whether the denominator is positive or not. If it is not positive multiply or divides numerator and denominator with -1 so that the denominator no longer remains negative.

Step 3: Determine the GCD of the absolute values of both numerator and Denominator.

Step 4: Divide the numerator and denominator with the GCD obtained in the earlier step. Thereafter, the rational number obtained is the standard form of the given rational number.

Solved Examples

1.  Determine whether the following Rational Numbers are in Standard Form or Not?

(i) -8/23 (ii) -13/-39

Solution:

-8/23 is said to be in Standard Form since both the numerator and denominator doesn’t have any common factors other than 1. In fact, the denominator is also positive. Thus, the given rational number -8/23 is said to be in its Standard Form.

-13/-39 is not in standard form since it has common factor 13 along with 1. Moreover, the denominator is not positive. Thus we can say the given rational number is not in standard form.

2. Express the Rational Number 18/45 in Standard Form?

Solution:

Given Rational Number 18/45

Check for the denominator in the given rational number. Since it is positive you need not do anything.

Later find the GCD of the absolute values of numerator 18, denominator 45

GCD(18, 45) = 9

Thus, to convert the given rational number 18/45 to standard form simply divide both the numerator and denominator by 9

18/45 = (18÷9)/(45÷9)

= 2/5

Therefore, 18/45 expressed in standard form is 2/5.

3. Find the Standard Form of 12/-18?

Solution:

Given Rational Number is 12/-18

Check for the denominator in the given Rational Number

Since the denominator, -18 has a negative sign multiply both numerator and denominator with -1 to make it positive.

12/-18 = 12*(-1)/-18*(-1)

= -12/18

Find the GCD of absolute values of both numerator and denominator

GCD(12, 18) = 6

To convert a given rational number to its standard form multiply and divide both numerator and denominator by 6.

-12/18 = ((-12)÷6)/(18÷6)

= -2/3

Thus, the standard form of Rational Number 12/-18 is -2/3.

4. Reduce 3/15 to Standard Form?

Solution:

Given Rational Numbers is 3/15

Since the denominator is positive you need not do anything to change it to positive.

Find the GCD of absolute values of numerator and denominator of the given rational number.

GCD(3, 15) = 3

Divide numerator and denominator with GCD obtained.

3/15 = (3÷3)/(15÷3)

= 1/5

Therefore, 3/15 Reduced to Standard Form is 1/5.

What is Rational Numbers

What are Rational Numbers?

If you are willing to know What are Rational Numbers and general representation of it have a look at the further modules. The number that can be expressed in the form of a fraction a/b where a, b are integers and the denominator b is non zero is called a rational number.

You can also say that a Rational Number is a number that can be expressed as the quotient of two integers having the condition where the divisor is non zero.

Numerator and Denominator: If a/b is a rational number then integer a is the numerator and b is the denominator.

Examples: 3/2, 8/5, -14/3, -11/5 are all Rational Numbers as they have integers in numerators and denominators and denominators are non zero.

Vedic Maths

Vedic Maths | 2 Second Maths Tricks

Vedic Maths: Vedic Maths book was written by Swami Bharati Krishna Tirtha, who was an Indian monk. It was first published in 1965. Veda is basically a Sanskrit word, which means Knowledge. It has a list of mental calculation techniques which are said to be present in the Vedas. This book is actually mentioned as Vedic Mathematics. This has been criticized by academics, who have also opposed its formation in the Indian school curriculum.

Vedic Maths Tricks

This Mathematics is a collection of Techniques/Sutras to solve arithmetical problems in an easy and faster way. It consists of 16 Sutras and 13 sub-sutras which can be used for questions involved in arithmetic, algebra, geometry, calculus, conic. The sutras are basically the formulas which we use in the mathematical calculation.

Vedic Maths book was previously included in the syllabus of Madhya Pradesh and Uttar Pradesh Board affiliated Schools. Some of the schools and organizations run by Hindu nationalist groups, also have included methods in their syllabus, even those groups which are outside India. The Hindu nationalists also tried to include these curricula in the NCERT books.

In earlier classes, we studied that during the period of dwelling in Sringeri Matha, Mysore Swami Bharati Krishna Tirtha did an arduous ascetic fervor for eight years. In the highest position of accomplishment, he got the perceptive vision of mathematics formulae mentioned in Vedas, the holiest scriptures and expressed his spiritual feelings in the form of Mantras (formulae).
These mantras were named Vedic Mathematical Formulae, which is exactly true. According to the scholars of Vedas, the knowledge of Vedas is beyond human power. The knowledge of Vedas cannot be obtained only by thinking or learning. It is an accomplish-feeling and expression it in the form of Mantras. In this perspective, the formulae formed by Swamiji are Vedic Mathematics Mantras.

Importance of Vedic Mathematics
To get a proper solution of any mathematical problems when a regular and inure practice of Vedic mathematical formulae is made, the concentration and memory is developed also a fierceness is seemed in his thinking and meditation. Due to freshness, simplicity and agreeability of Vedic mathematics create a feeling of inquisitiveness in the human heart.
The inquisitiveness makes him alert and aware and steps by step his inner soul begin to be vigilant. In fact, Vedic mathematics is a science of making someones inner should awake. This awareness of inner soul becomes the basis of the development of the human mind and his personality.

Important Definitions Related to Vedic Mathematics

  • Preceding Number (Purvena): The previous number is called the preceding number e.g., in 785, the Previous number of 5 is 8.
  • Param Mitra Number or Complementary Number: Two numbers having the sum of 10 are called Param Mitra number or complementary numbers.
    e.g.,Param Mitra number of 2 = 8
    Param Mitra number of 5 = 5
    Param Mitra number of 9 = 1
  • Disjunctor Number: The number which is subtracted from other numbers.
  • Detachable Number: The number from which a number is subtracted.
  • Formula Ekadhikena Purvene: To obtain the next number to a given number.
    A dot (.) is put on the given digit or number of ekadhik to show this sutra e.g., ekadhik of 15 = 15 + 1 = 16
    Ekadhik of digit 5 in 2536 = 2636.
  • Nikhilum Digit: Except unit digit, all the digits are called Nikhilam digit e.g., in 963, 9, 6 are Nikhilam digit.
  • Charam Digit: Unit digit of a number is called charam digit e,g., 3 is charam digit in 963.
  • Vinculum number (negative digits): The digit having bar above on it is called vinculum number. e.g., -5 = \(\bar { 5 }\).
  • Base: 10, 100, 1000, etc. are called base. These are multiple of 10.
  • Sub-base: Sub-base is the multiplier of base mostly it is a digit ends with zero, e.g., In 10 × 8, 10 is based and 8 is sub-base.
  • Deviation: When base or sub-base is subtracted from the given number the difference is called deviation.
    Deviation = Number – Base (or sub-base)
    If number > base/sub-base, then deviation will be +ve
    If number < base/sub-base, then deviation will be -ve.
    A deviation has as many digits as the number of zeros in the base.
  • Param Digit: Digit 9 is called Param digit or Brahm digit.
  • Whole Number: Ten is called the whole number.

Main Operations: Practice and Extension
Addition: In our previous class, we learned and practiced the addition of real numbers with the help of Sutra Ekadhikena Poorvena. In the practice, problems related to perfect numbers (i.e., Measurement units km-m) were solved. In fact, using the Sutra Ekadhikena Purvena all the problems of addition can be solved, measurement unit, money (rupees and paisa), weight (kilogram-gram) capacitance (liter-milliliter), time (hours, minutes, seconds), decimal fractions real numbers, distance (km, m, cm), etc.
Step 1: Informing column, according to the unit of measurement there is certain numbers in unit column. For example one rupee = 100 paise, so there will be two unit columns. 5 paisa will be written as paisa 05. Similarly, 1 kilometer = 1000 meters, so there will be three unit columns in problem and 84 meters will be written as 084 meters, so there will be three unit columns in problem and 84 meters will be written as 084 meters.
Step 2: Having completed forming the column, operation of addition is finished with the help of formula.
Step 3: When the sums are related to time whose unit is an hour, minute and second, the base in the first column (10) and in second 6 should be taken. In the column of an hour the base is always 10.
Example 1.
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) 65 gram and 85 grams will be written as 065 grams ans 085 grams.
(ii) Start adding from the top of unit column.
(iii) 5 + 5 = 10, so mark the more than, sign above 8, it comes before 5, remainder = 10 – 10 = 0
(iv) Write 0 + 2 + 5 = 7 at the place of answer.
(v) Keep it up.

Addition Orally (Sutra Ekadikena Poorvena + Soonyant Sankhya)
After some regular practice of above sutra, sums with large numbers can be solved rapidly or orally. Use of a round figure is a special method of ancient Indian Mathematics which is easier and very affected in the operation of addition. In the method, two digits at one and tens place are added in a special manner. Three digits at one’s tens and hundreds can also be added accordingly.
Method: Make one number of two in a round figure maintain its deficiency in the second given number. Add obtained new numbers. If die obtained sum result is more than 100, put an ekadhikena mark above it. Add the remainder into next number. At last, write final remainder as an answer. Revise this operation for the next columns. Look at the example given below:
Example 1.
Add 35 and 58.
Solution:
Convert these 58 into a round figure we need 2.
This deficiency will maintain in 35.
So 35 + 58 = 33 + (2 + 58) = 33 + 60 = 93
Example 2.
Add 19 and 65.
Solution:
19 + 65 = (19 + 1) + 64 = 20 + 64 = 84
Note: Similarly, many numbers can be added.
Example 3.
Add.
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) 98 + 89 = 98 + 2 + 87 = 100 + 87 = 187
On the digit 7 before 89 more than one mark.
(ii) Remainder 87 + 15 = 87 + 3 + 12 = 90 + 12 = 102
On the digit 7 before 15, more than mark.
(iii) Remainder 02 + 37 + 39 and 39 + 76 = 35 + 4 + 76 = 35 + 80 = 15 + 20 + 80 = 115
above the digit 9 before 76 more than one mark and 15 is put at the place of answer.
(iv) Now complete the operation as given above.

Example 4.
Add
Vedic Maths Addition and Subtraction Tricks 1
Hints
(i) 34 + 59 = 33 + 1 + 59 = 33 + 60 = 93
(ii) Remainder 93 + 32 = 93 + 7 + 25 = 100 + 25 = 125.
So more than one mark on digit 9 in third column.
(iii) Remainder 25 + 47 = 22 + 3 + 47 = 22 + 50 = 72
(iv) Remaining operation given above.

Subtraction:
In the previous class, we used two Vedic methods to solve the subtractions.
1. The method based on the Sutra Ekadhikena Poorvena Mitra Ank.
2. The method based on the Sutra Ekanyunen Poorvena Param Mitram,
Every problem of subtraction of measure unit of the real number can be solved with the help of the first method.
So, we consider again on this method. We know that two digits of a number are Param Mitra (close friend) to each other if their sum is equal to or more than ten.
The number from which a number is subtracted is called detachable number and the number which is subtracted is called disjunctor. The method can be cleared in an easy way with the help of examples given below.
Example 1.
Subtract by Vedic Method.
Vedic Maths Addition and Subtraction Tricks 1
Hint
(i) 3 cannot be subtracted from 0, so add 7 the Param Mitra Anka of 3, to 0. Write the sum below at the answer place and mark more than one sign on previous digit 6.
(ii) 6 = 7 cannot be subtracted from 0, so add the digit 3, the Param Mitra and of 7 into 0. Write it at the answer place and mark the sign of more than one on previous digit 2.
(iii) Write 8 – \(\dot { 2 }\) = 5, at the answer place, So the remainder = 537.
Example 2.
Subtract the following by Vedic Method:
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) Measurement unit in time column-wise base is different.
(ii) In the column of minute and second there will be two bases.
(a) Base in the unit column of both = 10
(b) Base in tens column of both = 6
(iii) Base in the column of hour = 10
(iv) The base of getting complement digits in the tens column of minute and second = 6 and in remaining base 10.
Hence, the answer is 25H 57min. 45sec.
Example 3.
Subtract by Vedic Method:
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) Arrange the column numbers in meter and centimeter.
(ii) Centimeter column: 6 cannot be subtracted from 5, thus add 4 the Param Mitra Ank of 6 to 5.
(iii) Write the sum = 9 at the answer place and mark more than one sign on 4 the pre dissociator digit.
(iv) \(\dot { 4 }\) = 5 cannot be subtracted from 3, so add 5 the param Mitra and to 3, the pre-dissociator digit.
(v) Write sum = 8 below and mark the sign of more than one on the pre-dissociator 5.
(vi) Write 7 – \(\dot { 5 }\) = 1 below
(vii) 7 cannot be subtracted from 6, so adding 6 + 3 = 9 below and mark the sign of more than one the pre-disjunctor 3.
(viii) Write 4 – \(\dot { 3 }\) = 0 below.
(ix) All the next operations will be done in the same way.
Remainder = 9 km 91 m 89 cm.

Multiplication:
In the previous class, we studied four methods of multiplication operations based on the Vedic Sutras. We should have the perfect practice of these sutras so that we might be able to choose a proper sutra for a proper solution of any type of problem-related to multiplication at once. Look at the example given below.
Example 1.
Multiply the following :
686 × 614 (Sutra Ekadhikena purvena)
Solution:
686 × 614
sum = 100 = 6 × 7 / 86 × 14 = 42 / 1204 = 421204
Hints:
(i) Sum of lost’digit = 86 + 14 = 100
(ii) Remaining nikhilam digits are mutually equal = 5
(iii) Four digits in right side = 1204.
Hence, 686 × 614 = 421204.

Example 2.
Choose the best Sutra to get an easy and rapid solution of588 × 512.
Solution (i):
May the Sutra Ekadhikena Poorvena be the best formula for solving it?
The sum of the digits at the place of ones and tens = 88 + 12 = 100 and two Nikhilam digits are 5 each. So the Sutra of effective here.
According to the Sutra
588 × 512 = 5 × 6 / 88 × 12 (Four digits in RHS)
= 301056
Solution (ii): Justification of the Sutra Nikhilam Upadhara
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) Base = 100
(ii) Sub-base = 100 × 5
(iii) Sub-base digit = 5
(iv) Differences = +88 and +12
(v) Two digits in RHS and the Sutra is effective
Solution (iii): The Sutra Ekanayunena Purvena cannot be applied for finding the solution of 588 × 512, as there is not a digit of 9 in both numbers.
Solution (iv): 588 × 512 can be solved by the Sutra Urdhava-tiryagbhyam
There are three columns in the problem. So five groups will be formed i.e., having obtained five products, they will be written in a special manner and then added.
Vedic Maths Addition and Subtraction Tricks 1
Result 1. Observing the first, second and fourth solutions it is certain that 588 × 512 = 301056.
2. In the first solution, we get the result easier, therefore Sutra Ekadhikena Poorvena is the best.

Example 3.
Which of the Sutra is the best to solve 842 × 858?
Solution:
(i) In this problem the Sutra Ekadhikena Poorvena is not effective as the product of RHS i.e., 42 × 58 cannot be obtained easily.
(ii) The Sutra Nikhilam Adhara cannot be effective here as if the base 1000 is considered, then the difference respectively will be -158 and -142. The Sutra Nikhilam-Upadhara is not effective here as Upadhar (Sub-base) = 800, we get the difference 42 and 58 respectively.
(iii) The Sutra Ekanyunena Poorvena is not effective here.
(iv) The Sutra Urdhva-tiryagbhyam is effective and the best here. If the number is large, the calculation is difficult. So we can think over this new formula.
(v) The New option: To get the solution of 842 × 858, two Sutras are used respectively, first the Sutra Ekadhikena Poorvena and then the Sutra Urdhva-tiryagbhyam.
Vedic Maths Addition and Subtraction Tricks 1
(v) Extension of the Operation of Multiplication (Sutra Urdhvatiryagbhyam + Viloknam)
Using the formulae Urdhva-tiryagbhyam and vinculum, the product of two larger number can be obtained id easily. Using the vinculum siitra first of all. The digits more than 5 are converted into smaller digits i.e., (0, 1, 2, 3, 4, 5) after that the product is calculated with the Sutra Urdhva-tiryagbhyam and at last obtained product, including negative sign is again turned into dying normal digits. this method can be cleared by observing the examples given below.
Example 1.
Solve 842 × 858
Vedic Maths Addition and Subtraction Tricks 1
Hints
(i) Convert greater digits into smaller ones.
(ii) Multiple by Urdhva-tiryagbhyam Sutra.
(iii) Turn the obtained negative signs of the product in the normal digits by the method of Nikhilam.

Example 2.
Find the value of 966 × 973.
Vedic Maths Addition and Subtraction Tricks 1
Note: Using the Sutra Urdhva-tiryagbhyam, we should try to get the product of large number orally and express it into one line.

Division:
In the previous class, the following three methods were used for division in detail:
(a) Sutra Nikhilam
The Nikhilam Sutra based method is effective only when the digits of divisor are greater than 5 and the complementary number of division to the respect of the base 10 or the power of 10 is known. In this method, the main operation is performed by the complementary number.

(b) Sutra Paravartya Yojayet
Sutra Paravartya Yojayet will be effective only when the digits of divisor are smaller than 5 or can be put also the digit from the left-hand side is 1 or can be brought and the base = 10, or the difference of the divisor comparably the power of 10 (not sub-base). Only this method out of three can be used for the division in algebra.

(c) Sutra Urdhva-Tiryabhyam
By the method of dhvanjank based on the Urdhva-tiryagbhyam, every type of problem-related to division can be solved. In this method, it is most important to choose the Mukhyank and dhvajank properly. The dhvanjank may have so many digits as you wish. There may be more than one digits in Mukhyank. As many digits must be put in the dividend at the place of ones in die third section as there are in the dhvajank and remaining in the middle section. We shall clear the method with the help of the following examples:
Example 1.
Solve by the dhvajank method
989765 ÷ 87
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) Divisior = 87, Mukhyank = 8 and dhvajank = 7
(ii) In third column one digit of dividend = 5.
(iii) 9 ÷ 8, first digit of quotient = 1 and remainder = 1.
(iv) New dividend = 18, the corrected dividend.
(v) 11 ÷ 8, second digit of quotient = 1, remainder = 3.
(vi) New dividend = 37, the corrected dividend.
(vii) 30 ÷ 8, third digit of quotient = 3, remainder = 6.
(viii) New dividend = 66, the corrected dividend.
(ix) 45 ÷ 8, forth digit of quotient = 5, remainder = 5.
(x) New dividend = 55, the corrected dividend or the final remainder
Quotient = 1135, remainder = 20

Example 2.
Solve: 13579 ÷ 975 (dhvajank method)
Solution: 13579 ÷ 975
Vedic Maths Addition and Subtraction Tricks 1
Hint:
(i) 13 ÷ 9, first digit of quotient = 1, remainder = 4.
(ii) New dividend = 45,corrected dividend = 45 – 1 × 7 = 38
(iii) 38 ÷ 9, second digit of quotient = 4, remainder = 2.
(iv) New dividend = 27
Corrected dividend = 27 – (4 × 7 + 1 × 5) = 27 – 33 = -6
Since, we get the corrected dividend is negative so the second digit of quotient must be 3 instead of 4.
This is why the terms (iii) and (iv) are rejectable.
(v) Again 38 ÷ 9, the second digit of quotient = 3, remainder = 11
(vi) New dividend = 1179. So the corrected dividend or the final dividend or final remainder.
= 1179 – (3 × 7 + 1 × 5) × 10 – 3 × 5 = 1179 – 260 – 15 = 904
Hence, the quotient = 13 and remainder = 904

Example 3.
21015 ÷ 879 (dhvanjank method)
Solution:
Divisor = 897, Mukhyank = 8 and dhvanjank = 79.
Since there are larger digits in the dhvanjank, so the divisor 879 will be converted into smaller digits comparatively by Nikhilam (Vinculum) Method
Vedic Maths Addition and Subtraction Tricks 1
Hints:
(i) 21 ÷ 9, first digit of quotient = 2, remainder = 3.
(ii) New dividend = 309, corrected dividend = 30 – 2 × \(\bar { 2 }\) = 34
(iii) 34 ÷ 9, second digit of quotient = 3, remainder = 7.
(iv) New dividend = 715, corrected dividend or final remainder
= 715 – (3 × \(\bar { 2 }\) + 2 × \(\bar { 1 }\))10 – 3 × \(\bar { 1 }\)
= 715 + 80 + 3 = 798
Hence, quotient = 23, remainder = 798.

Example 4.
7453 ÷ 79 (dhwanjank method)
Hints:
(i) Divisor 79 = 8 \(\bar { 1 }\), mukhyank = 8, dhvanjank = \(\bar { 1 }\)
(ii) 74 ÷ 8, first digit of quotient = 9, remainder = 2
(iii) New dividend = 25, corrected dividend = 25 + 9 = 34
(iv) 34 ÷ 8, second digit of quotient = 4, remainder = 2.
(v) New dividend or final remainder = 23 + 4 = 27.
Hence quotient = 94, remainder = 27
Note:
1. See the construct (iii)
New dividend = 25, corrected dividend = 25 – \(\bar { 9 }\) × 1 = 25 + 9 = 34.
New dividend + pre digit of the quotient.
2. If 9 is at the place of ones in the divisor, then the corrected dividend = New dividend = pre quotient digit may be taken.
3. In the problem in the which the one’s place unit, of divisor, is 1, the corrected dividend = new dividend + the digit of pre quotient is taken.
4. There is no need to write the hints for the two types of above problems.

Example 5.
Solve 43758972 ÷ 81 (dhvajank method)
Solution:
43758972 ÷ 81
Vedic Maths Addition and Subtraction Tricks 1

Addition and Subtraction Vedic Maths Tricks

Vedic Maths Addition and Subtraction Tricks 1

Vedic Maths Addition and Subtraction Tricks 2 Vedic Maths Addition and Subtraction Tricks 3 Vedic Maths Addition and Subtraction Tricks 4 Vedic Maths Addition and Subtraction Tricks 5 Vedic Maths Addition and Subtraction Tricks 6

Vedic Maths Advantages and Uses

Once the student understands the system of mental mathematics, they will become more creative and start thinking logically. This maths is very flexible for them. The students can easily play with numbers with the help of this system.

Regular mathematical methods are sometimes complex and time-consuming. But if use Vedic Mathematic’s Procedure and Techniques, some of the calculations such as, sets of given data,  can be done very fast. Some of the more useful advantages of this Mathematics are;

  • Its more than 1700% times faster than General Math. Thus, it could be considered as the World’s Fastest.
    It helps a child to lose the Math fear from his mind. Usually, students are scared of doing mathematical calculation because of the logic they have to use to solve it. Thus, this ancient Maths will help to solve the problems in the easiest way.
    It helps to increase the thinking capability and intelligence, along with a sharpening of the mind.
  • Helps in increasing the speed and giving accurate answers.
  • Students feel very confident about this subject, after improving their memory power.
  • Students get interested more in numbers, they just have to apply their skills and have knowledge of tables to learn this.
  • These maths calculation are of great use while preparing for competitive exams.

Vedic Maths Sutras

As mentioned before, it consists of 16 sutras. Below are the names of the Sutras and Upa sutras with their meaning and corollary.

Name Upa Sutra Meaning Corollary
Ekadhikena Purvena Anurupyena By one more than the previous one Proportionately
Nikhilam Navatashcaramam Dashatah Sisyate Sesasamjnah All from 9 and the last from 10 The Remainder Remains Constant
Urdhva-Tiryagbyham Adyamadyenantyamantyena Vertically and crosswise First by the First and the Last by the Last
Paraavartya Yojayet Kevalaih Saptakam Gunyat Transpose and adjust For 7 the Multiplicand is 143
Shunyam Saamyasamuccaye Vestanam When the sum is the same that sum is zero By Osculation
Anurupye Shunyamanyat Yavadunam Tavadunam If one is in ratio, the other is zero Lessen by the Deficiency
Sankalana-vyavakalanabhyam Yavadunam Tavadunikritya Varga Yojayet By addition and by subtraction Whatever the Deficiency lessen by that amount and set up the Square of the Deficiency
Puranapuranabyham Antyayordashake’pi By the completion or non-completion Last Totalling 10
Chalana-Kalanabyham Antyayoreva Differences and Similarities Only the Last Terms
Yaavadunam Samuccayagunitah Whatever the extent of its deficiency The Sum of the Products
Vyashtisamanstih Lopanasthapanabhyam Part and Whole By Alternate Elimination and Retention
Shesanyankena Charamena Vilokanam The remainders by the last digit By Mere Observation
Sopaantyadvayamantyam Gunitasamuccayah Samuccayagunitah The ultimate and twice the penultimate The Product of the Sum is the Sum of the Products
Ekanyunena Purvena Dhvajanka By one less than the previous one. On the Flag
Gunitasamuchyah Dwandwa Yogiji The product of the sum is equal to the sum of the product. –
Gunakasamuchyah Adyam Antyam Madhyam The factors of the sum are equal to the sum of the factors –

We hope the detailed article on Vedic Maths is helpful. If you have any doubt regarding this article or Vedic Maths, drop your comments in the comment section below and we will get back to you as soon as possible.

FAQs on Vedic Maths

1. What is Vedic Mathematics?

Vedic Maths is the world’s fastest mental maths system. It helps you calculate faster.

2. How useful is Vedic Maths?

Vedic Maths improves mentality and promotes creativity. The ease and simplicity of Vedic Mathematics mean that calculations can be carried out mentally. All these features of Vedic math encourage students to be creative in doing their math.

3. What are the best ways to learn Vedic Maths?

Learn about the Vedic Maths Tricks, Primary Definitions, and the importance of it using the direct links available on our page. You can view or download and use them as a reference during your preparation.

4. Which are good books to learn Vedic Mathematics?

Aspirants can look for some of the best books to learn on Vedic Mathematics by accessing our page.

 

Lowest Form of a Rational Number

Rational Numbers are said to be in the lowest form if both the numerator and denominator have no other common factor other than 1. In other words, we can say that a Rational Number a/b is said to be in its simplest form if the HCF of a, b is 1 and a, b are relatively prime. To help you understand the concept better we have illustrated a few examples of how to convert Rational Numbers to their respective lowest form.

Example: 5/3 is a Rational Number in the Lowest Form or Simplest Form as it has no other common factors other than 1 Whereas, 14/21 is not in the simplest form as it has common factor 7 along with 1. Thus, it is not in the simplest form.

How to Convert a Rational Number to Lowest or Simplest Form?

Go through the below steps to change a Rational Number into its Lowest or simplest form. The Guidelines are along the lines

Step 1: Firstly, note down the rational number a/b given.

Step 2: Find the HCF of a, b

Step 3: If you get the HCF(a, b) i.e. k = 1 then they are in the simplest form.

Step 4: If at all the HCF(a,b) i.e. k ≠ 1 the simply divide both the numerator and denominator by k i.e. (a÷k)/(b÷k) to obtain the lowest form of rational number a/b. 

Solved Examples

1.  Determine whether the following rational numbers are in their lowest form or not?

(i)14/81 (ii) 72/24

Solution:

1/81 is in its lowest form as the HCF is 1 i.e. both the numerator and denominator have no common factors other than 1.

(ii) 72/24

72 = 2 × 2 × 2 × 3 × 3, 24 = 2 × 2 × 2 × 3

HCF of 72, 24 is 24

Therefore, the rational number 72/24 is not in the lowest form.

2. Express 45/30 in its simplest form?

Solution: 

We have 45/30

Find the HCF of 45, 30 initially

HCF(45, 30) = 15

As the HCF is not equal to 1 simply divide the numerator and denominator of the given rational number with 15.

45/30 = (45÷15)/(30÷15) = 3/2

Therefore, 45/30 in its simplest form is 3/2.

3. Express the Rational Number 36/24 into its Lowest Form?

Solution:

Given Rational Number 36/24

Figure out the HCF(36, 24) at first

HCF(36, 24) = 12

Since HCF is not equal to 1 divide both the numerator and denominator with the HCF obtained.

36/24 = (36÷12)/ (24÷12)

= 3/2

Therefore, 36/24 expressed into its lowest form is 3/2.

4. Reduce 3/15 to its Lowest Form?

Solution:

Given Rational Number is 3/15

Find out the HCF(3,15)

HCF(3,15) = 3

As the HCF isn’t equal to 1 divide the numerator and denominator of rational number with HCF to change it to the lowest form.

3/15 = (3÷3)/(15÷3)

= 1/5

Properties of Rational Numbers

A Rational Number is a number that can be written in the form of p/q  where p, q are integers, and q ≠ 0. You can learn about the General Properties of Rational Numbers like Closure, Commutative, Associative, Distributive, Identity, Inverse, etc. here. Not just the regular properties we have all listed all the properties that we know regarding Rational Numbers.

Closure Property

For two rational numbers x, y the addition, subtraction, multiplication results always yield a rational number. The Closure Property isn’t applicable for the division as division by zero isn’t defined. In other words, we can say that closure property is applicable for division too other than zero.

4/7 + 2/3 =26/21

4/3 – 2/4 = 6/12

3/5. 2/3 = 6/15

Commutative Property

Considering two rational numbers x, y the addition and multiplication are always commutative. Subtraction doesn’t obey commutative property. You can get a clear idea of this property by having a look at the solved examples.

Commutative Law of Addition: x+y = y+x Ex: 1/3+2/3 = 3/3

Commutative Law of Multiplication: x.y = y.x Ex: 1/2.2/3 =2/3.1/2 =2/6

Subtraction x-y≠y-x Ex: 4/3-1/3 = 3/3 whereas 1/3-4/3=-3/3

Division isn’t commutative x/y ≠y/x Ex: 3/9÷1/2=6/9 whereas 1/2 ÷3/9 =9/6

Associative Property

Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z

whereas for Multiplication x(yz)=(xy)z

Ex: 1/3 + (1/4 + 3/3) = (1/3+ 1/4) + 3/3

⇒19/12 =19/12

Distributive Property

Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.

Ex: 1/3.(1/4+2/5) =(1/3.1/4)+(1/3.2/5)

1/3.(17/20)= 1/12+2/10

17/60 =17/60

Thus, L.H.S = R.H.S

Identity and Inverse Properties of Rational Numbers

Identity Property: We know 0 is called Additive Identity and 1 is called Multiplicative Identity of Rational Numbers.

Ex: 1/4+0 = 1/4(Additive Identity)

5/3.1 = 5/3(Multiplicative Identity)

Inverse Property: For a Rational Number x/y additive inverse is -x/y and multiplicative inverse is y/x.

Ex: Additive Inverse of 2/3 is -2/3

Multiplicative Inverse of 4/5 is 5/4

There are few other properties that you need to be aware of Rational Numbers and they are explained below.

Property 1:

If a/b is a rational number and m is a non-zero integer then a/b =(a*m)/(b*m).

In other words, we can say that the rational number remains unaltered if we multiply both the numerator and denominator with the same integer.

Ex: 2/3 = 2*2/3*2 = 4/6, 2*3/3*3 = 6/9, 2*4/3*4 = 8/12….

Property 2:

If a/b is a rational number and m is a common divisor then a/b = (a÷m)/(b÷m)

On dividing the numerator and denominator of a rational number with a common divisor the rational number remains unchanged.

Ex: 36/42 =36÷6/42÷6 = 6/7

Property 3:

Consider a/b, c/d to be two rational numbers.

Then a/b = c/d ⇒ a*d = b*c

Ex: 2/4 =4/8 ⇒ 2.8=4.4

Property 4:

For each and every Rational Number n, any of the following conditions hold true.

(i) n>0, (ii) n=0, (iii) n<0

Ex: 3/4 is greater than 0.

0/5 is equal to 0.

-3/4 is less than 0.

Property 5:

For any two rational numbers a, b any one condition is true

(i) a>b, (ii) a=b, (iii) a<b

Ex: 2/3 and 2/5 are two rational numbers and 2/3 is greater than 2/5

If 4/8 and 8/16 are two rational numbers then 4/8 = 8/16

If -4/7 and 3/4 are two rational numbers then -4/7 is less than 3/4

Property 6:

In the case of three rational numbers a > b, b > c then a>c

If 4/5, 16/30, -8/15 are three rational numbers then 4/5 >16/30 and 16/30 is greater than -8/15 then 4/5 is also greater than -8/15.

Is Every Rational Number a Natural Number?

Every Natural Number is a Rational Number but it’s not the same in the case of Rational Numbers. There is no such thing when it comes to a Rational Number it may or may not be a Natural Number.

We know that 1 = 1/1, 2 = 2/1, ….

In fact, we can say a natural number n can be expressed as n = n/1 which is nothing but the quotient of two integers. Therefore, every natural number is a Rational Number.

On the other hand, 5/2, 3/5, 2/7, 4/20, etc. are all Rational Numbers but aren’t natural numbers.

Therefore, every natural number is a Rational Number but a Rational Number need not be a Natural Number.

Determine Whether the Following Rational Numbers are Natural Numbers or Not

(i) 5/2

5/ 2 is not a natural number.

(ii) 8/2

8/2 is a natural number as on simplifying it we get the result as 4 which is a natural number.

(iii) -20/5

-20/5 isn’t a natural number as on further simplifying we get the result -4 which is an integer but not a natural number.

(iv) -6/-3

-6/-3 is a natural number as we will get the result 2 on simplification which is a natural number.

(v) 1/5

1/5 is not a natural number.

(vi) 0/3

0/3 is not a natural number since 0/3 =0 and 0 is not a natural number.

(vii) 5/5

5/5 is a natural number on simplifying to its lowest term we get 1/1 = 1 and 1 is a natural number.

(viii) 27/45

27/45 is not a natural number as we get 3/5 on reducing to its lowest term which is a rational number but not a natural number.

Thus, by looking after the instances above we can state that Not every Rational Number is a Natural Number.

Properties of Addition of Rational Numbers

Learn the Properties of Rational Numbers such as Closure Property, Commutative Property, Identity Property, Associative Property, Additive Inverse Property, etc. We tried explaining each and every Property of Rational Numbers Addition in the following sections. Check them and learn the concepts easily. To help you understand the concept, even more, better we have provided a few examples.

Closure Property of Addition of Rational Numbers

Closure Property is applicable for the Addition Operation of Rational Numbers. The Sum of Two Rational Numbers always yields in a Rational Number. Let a/b, c/d be two rational numbers then (a/b+c/d) is also a Rational Number.

Examples

(i) Consider the Rational Numbers 5/4 and 1/3

= (5/4+1/3)

= (5*3 +1*4)/12

= (15+4)/12

= 19/12

Therefore, the Sum of Rational Numbers 5/4 and 1/3 i.e. 19/12 is also a Rational Number.

(ii) Consider the Rational Numbers -4/3 and 2/5

= -4/3+2/5

= (-4*5+2*3)/15

= (-20+6)/15

= -14/15 is also a Rational Number.

Commutative Property of Addition of Rational Numbers

Commutative Property is applicable for the Addition Operation of Rational Numbers. Two Rational Numbers can be added in any order. Let us consider two rational numbers a/b, c/d then we have

(a/b+c/d) = (c/d+a/b)

Examples

(i) 1/3+4/5

= (5+12)/15

= 17/15

and 4/5+1/3

= (12+5)/15

= 17/15

Therefore, (1/3+4/5) = (4/5+1/3).

(ii) -1/2+3/2

= (-1+3)/2

= 2/2

and 3/2+(-1/2)

= (3-1)/2

= 2/2

Therefore, (-1/2+3/2) = (3/2+-1/2)

Associative Property of Addition of Rational Numbers

While adding Three Rational Numbers you can group them in any order. Let us consider three Rational Numbers a/b, c/d, e/f we have

(a/b+c/d)+e/f = a/b+(c/d+e/f)

Example

Consider Three Rational Numbers 1/2, 3/4 and 5/6 then

(1/2+3/4)+5/6 = (2+3)/4+5/6

= 5/4+5/6

= (15+10)/12

= 25/12 and

1/2+(3/4+5/6) = 1/2+(9+10)/12

= 1/2+19/12

= (6+19)/12

=25 /12

Therefore, (1/2+3/4)+5/6 = 1/2+(3/4+5/6)

Additive Identity Property of Addition of Rational Numbers

0 is a Rational Number and any Rational Number added to 0 results in a Rational Number.

For every Rational Number a/b (a/b+0)=(0+1/b)= a/b and 0 is called the Additive Identity for Rationals.

Example

(i) (4/5+0) = (4/5+0/5) =(4+0)/5 =4/5 and similarly (0+4/5) = (0/5+4/5) = (0+4)/5 = 4/5

Therefore, (4/5+0) = (0+4/5) = 4/5

(ii)(-1/3+0) =(-1/3+0/3) =(-1+0)/3 = -1/3 and similarly (0-1/3) = (0/3-1/3) =(0-1)/3 = -1/3

Therefore, (-1/3+0) =(0+-1/3) = -1/3

Additive Inverse Property of Addition of Rational Numbers

For every Rational Number a/b there exists a Rational Number -a/b such that (a/b+-a/b)=0 and (-a/b+a/b)=0

Thus, (a/b+-a/b) = (-a/b+a/b) = 0

-a/b is called the Additive Inverse of a/b

Example

(4/3+-4/3) = (4+(-4))/3 = 0/3 = 0

Similarly, (-4/3+4/3) = (-4+4)/3 = 0/3 = 0

Thus, 4/3 and -4/3 are additive inverse of each other.