Paravartya in solving simple equations:

Paravartya in solving simple equations:

'Paravartya yojayet' means 'transpose and apply'. According to the rule invariable change its sign with every change of side from left to right, (+) becomes (-) and; and (X) becomes (÷). Further it can be extended from numerator to denominator in the concerned problems.

प्रत्येक पक्षान्तरण में गणितीय राशियों के चिन्ह परिवर्ति‍त होते हैं। इस प्रकार (+) चिन्ह (-) हो जाता हैं व (-) चिन्ह (+) हो जाता हैं, (x) का (÷) व (÷) का (x) हो जाता हैं।

Application 1: If ax + b = cx + d.

By paravartya, we get-

Example: 4x + 3 = 2x + 9

Here a =4, b = 3, c = 2, d = 9

Application 2: If (x + a) (x +b) = (x +c) (x +d).

By paravartya, we get -

Example: (x + 7) (x + 9) = (x - 8) (x - 11).

Here a =7, b = 9, c = - 8, d = -11

Application 3: If

By paravartya, we get-

Example:

Application (4): If

By paravartya we get-

Example:

Application (5): If

Example:

Application (6): If

Example:

Simple equations:

By Paravartya sutra we can derive the values of x and y. which are given in two simple equations.

Example:

2x + 3y = 13,

4x + 5y = 23.

1.      x का मान ज्ञात करने के लिए दोनों समीकरण के y के गुणक तथा अचर राशियों का बज्र गुणा करते हैं, तथा बज्र गुणा से प्राप्त राशियों को घटाते हैं, प्राप्त संख्या x के लिए अंश के रुप में प्रयुक्त होती हैं।

2x + 3y = 13

4x + 5y = 23

“X” के लिए अंश

= 3 x 23 – 5 x 13

= 69 – 65 = 4

2.      दोनों समीकरणों के x तथा y के गुणक का बज्र गुणा कर घटाने पर प्राप्त संख्या x के लिए हर के रुप में प्रयुक्त होती हैं।

“X” के लिए हर

= (3 x 4) – (2 x 5)

= 12 – 10 = 2       

अतः X = 4 ÷ 2 = 2

3.      y का मान ज्ञात करने के लिए दोनों समीकरण के x के गुणक तथा अचर राशियों का बज्र गुणा करते हैं, तथा बज्र गुणा से प्राप्त राशियों को घटाते हैं, प्राप्त संख्या y के लिए अंश के रुप में प्रयुक्त होती हैं।

“Y” के लिए अंश

= (13 x 4) – (23 x 2)

= 52 – 46 = 6

4.      y के लिए हर = 2; जो पद 2 से प्राप्त हुआ।

अतः y = 6÷2 = 3

अतः समीकरण में, x = 2 तथा y = 3

Adyamadyena-Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)

Adyamadyena-Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)

The Sutra “Adyamadyena-Antyamantyena” means “the first by the first and the last by the last”.

प्रथम को प्रथम के द्वारा तथा अन्तिम को अन्तिम के द्वारा।

Area of rectangle:

Example:

Find out the area of a rectangle whose length and breadth are respectively 5 ft.2 inches and 4 ft.5 inches.

Generally we continue the problem like this.

Area    = Length X Breadth

= 5’ 2" X 4’ 5"               (Since 1’ = 12")

= (5 X 12 + 2) (4 X 12 + 5) conversion in to single unit

= 62" X 53" = 3286 Sq. inches.

Since 1 sq. ft. =12 X 12 = 144 sq.inches

We have area

3286 /144 = Quotient is 22 and Remainder is 118.

Area of rectangle is 22 Sq. ft 118 Sq. inches.

Mental argumentation:

It is interesting to know the mental argumentation. It goes in his mind like this

    5’     2"

    4’     5"

First by first: 5’ X 4’ = 20 sq. ft.

Last by last: 2" X 5" = 10 sq. in.

Now cross wise 5 X 5 + 4 x 2 = 25 +8 = 33.

Adjust units  to left as  33 = 2 X 12 +9 , 2 twelve's as 2 square feet make the first 20+2 = 22 sq. ft ; 9 left becomes 9 x 12 square inches and go towards right  9 x 12 = 108 sq. in. gives 108+10= 118 sq.inch.

We got area in some sort of 22 sq ft and 128 sq. inches.

By Vedic principles "the first by first and the last by last"

5’ 2" can be treated as 5a + 2 and 4’ 5" as 4a + 5,

Where a= 1ft. = 12 inch and a² = 1 sq. ft = 144 sq. inch.

= (5a + 2) (4a + 5)

= 20a² + 25a + 8a + 10

= 20a² + 33a + 10

= 20a² + (24a+9a) + 10

= 20a²+ (2a+9) a + 10     writing 33 = 2X12 +9

= 22a²+ 9a + 10

= 22 sq. ft. + 9X12 sq. inch + 10 sq. inches

= 22 sq. ft. + 108 sq. inch + 10 sq. inches

= 22 sq. ft. + 118 sq. inch

Factorization of quadratics:

By Vedic process two sub-sutras are used to factorizing a quadratic.

(a) Anurupyena   (b) Adyamadyena-Antyamantyena

The usual procedure of factorizing a quadratic is as follows:

= 2 a² + 9a + 10

= 2 a² + 4a + 5a + 10

= 2a (a + 2) + 5 (a + 2)

= (a + 2) (2a + 5)

But by mental process, we can get the result immediately. The steps are as follows.

1.      Split the middle coefficient in to two such parts that the ratio of the first coefficient to the first part is the same as the ratio of the second part to the last coefficient. Thus we split the coefficient of middle term of 2 a² + 9a + 10 i.e. 9 in to two such parts 4 and 5 such that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2:4 and the ratio of the second pat to the last coefficient, i.e. 5: 10 are the same. It is clear that 2:4 = 5:10. Hence such split is valid. Now the ratio 2: 4 = 5: 10 = 1:2 give one factor (a+2). 

2.      Second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of the factor. i.e. the second factor is

2 a² + 9a + 10 = (a + 2) (2a + 5)

Formula 3

Anurupyena (आनुरूप्येण)

The upa-Sutra 'Anurupyena' means 'proportionality' or 'similarly'.

अनुपातों से।

इस सूत्र के उपयोग से आनुपातिक गुणन या भाग किया जाता हैं। जब संख्याऐं सैद्धान्तिक आधार 100 से काफी दूर हो तो क्रियात्मक आधार उपयोग में लाया जाता हैं।

This Sutra is highly useful to find products of two numbers when both of them are near the Common bases like 50, 100 etc (multiples of powers of 10).

Example: 47 X 42

As per the previous methods, if we select 100 as base we get

47     -53

42     -58

This is much more difficult. Now by ‘Anurupyena’ we take a different working base through we can solve the problem. Take the nearest higher multiple of 10. In this case it is 50.

Treat it as 100 / 2 = 50.

1.      We choose the working base. Working base is 100 / 2 = 50

2.      Write the numbers one below the other

47

42

__

3.      Write the differences from 50 against each number on right side

47     -03

42     -08

________

4.      Write cross-subtraction or cross- addition as the case may be under the line drawn. Multiply the differences and write the product in the left side of the answer.

5.      Since base is 100 / 2 = 50, 39 in the answer represent 39X50.

Hence divide 39 by 2 (because 50 = 100 / 2). Thus 39 ÷ 2 gives 19½ where 19 is quotient and ½ is remainder. This ½, as Reminder gives 50; making the L.H.S of the answer, 24 + 50 = 74 or (½ x 100 + 28) i.e. R.H.S. 19 and L.H.S. 74 together give the answer 1974.

Formula 2

s 100. Now follow the rules, 92 is 8 less than base 100. And 86 is 14 less than the same base 100. Hence 8 and 14 are called deviations from the base.

ii.   When both the numbers are higher than the base.

Here the deviation is positive as the numbers are higher then base.

We consider 03x07=21. This is done because, we need to consider two digits in deviation as it the base 100 has two zeros. If the deviation is near 1000 then we need to consider 3 digits in the deviation (e.g., 004 and not just 4).

iii. One number is more and the other is less than the base.

In this situation one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted.

12/ (-8) =112 how?

12/ (-8) should be read as 'one two, eight bar'. Here 'one' and ' two ' are in normal form. ‘Eight' is in complement form (परम मित्र रुप में). So, when we bring a carry from normal form to complement form, '12' becomes '11' and 'eight bar' becomes '2'( complement of 8). Hence 12/ (-8) =112.

Formula 1

Ekadhikena Purvena (एकाधिकेन पूर्वेण)

The Sutra means: “By one more than the previous one”.

पहले से एक अधिक के द्वारा।

1. This sutra is useful to the ‘squaring of numbers ending in 5’.

Example: 25².

For the number 25, the last digit is 5 and the 'previous' digit is 2. According to formula 'One more than the previous one', that is, 2+1=3. The Sutra gives the procedure 'to multiply the previous digit 2 (by one more than itself) by 3. It becomes the L.H.S (left hand side) of the result, that is, 2 X 3 = 6. The R.H.S (right hand side) of the result is 5², that is, 25.

252 = 2 X 3 / 25 = 6/25=625.

152 = 1 X (1+1) /25 =225;

952 = 9 X 10/25 = 9025;

135² = 13 X 14/25 = 18225;

2. Vulgar fractions whose denominators are numbers ending in Nine

We take examples of 1 / a9, where a = 1, 2………… In the conversion of vulgar fractions into recurring decimals, Ekadhika process can be effectively used both in division and multiplication.

Division Method:  Value of 1 / 19.

The numbers of decimal places before repetition is the difference of numerator and denominator, 19 -1=18 places. For the denominator 19, the purva (previous) is 1. Hence Ekadhikena purva (one more than the previous) is 1 + 1 = 2.

1.      0.₁0 (Divide numerator 1 by 20, 0 times, 1 remainder)

2.      0.0₀5 (Divide 10 by 2, 5 times, No remainder)

3.      0.05₁2 (Divide 5 by 2, 2 times, 1 remainder)

4.      0.0526 (Divide ₁2 or12 by 2, 6 times, No remainder)

5.      0.05263 (Divide 6 by 2, 3 times, No remainder)

6.      0.05263₁1(Divide 3 by 2, 1 time, 1 remainder)

7.      0.052631₁5 (Divide ₁1 or 11 by 25 times, 1 remainder)

8.      0.0526315₁7 (Divide ₁5 or 15 by 2, 7 times, 1 remainder)

9.      0.05263157₁8 (Divide ₁7 or 17 by 2, 8 times, 1 remainder)

10.       0.0526315789 (Divide ₁8 or 18 by 2, 9 times, No remainder)

11.       0.0526315789₁4 (Divide 9 by 2, 4 times, 1 remainder)

12.       0.052631578947 (Divide ₁4 or 14 by 2, 7 times, No remainder)

13.       0.052631578947₁3 (Divide 7 by 2, 3 times, 1 remainder)

14.       0.0526315789473₁6 (Divide ₁3 or 13 by 2, 6 times, 1 remainder)

15.       0.052631578947368 (Divide ₁6 or 16 by 2, 8 times, No remainder)

16.       0.0526315789473684 (Divide 8 by 2, 4 times, No remainder)

17.       0.05263157894736842 (Divide 4 by 2, 2 times, No remainder)

18.       0.052631578947368421 (Divide 2 by 2, 1 time, No remainder)

0 

Multiplication Method: Value of 1 / 19

For 1 / 19, 'previous' of 19 is 1. And one more than of it, is 1 + 1 = 2. Therefore 2 is the multiplier for the conversion. We write the last digit in the numerator (अंश) as 1 and follow the steps leftwards.

1.      1

2.      21(multiply 1 by 2, put to left)

3.      421(multiply 2 by 2, put to left)

4.      8421(multiply 4 by 2, put to left)

5.      ₁68421 (multiply 8 by 2=16, 1 carried over, 6 put to left)

6.      ₁368421 (6 X 2 =12, +1 = 13, 1 carried over, 3 put to left)

7.      7368421 (3 X 2, = 6 +1 = 7, put to left)

8.      ₁47368421 (7 X 2 =14, 1 carried over, 4 put to left)

9.      947368421 (4 X 2, = 8, +1 = 9, put to left)

10.       ₁8947368421(9 X 2 =18, 1 carried over, 8 put to left)

11.       ₁78947368421(8 X 2 =16,+1=17, 1 carried over, 7 put to left)

12.       ₁578947368421(7 X 2 =14,+1=15, 1 carried over, 5 put to left)

13.       ₁1578947368421(5 X 2 =10,+1=11,1 carried over, 1 put to left)

14.       31578947368421(1 X 2 =2,+1=3, 3 put to left)

15.       631578947368421(3 X 2 =6, 6 put to left)

16.       ₁2631578947368421(6 X 2 =12, 1 carried over, 2 put to left)

17.       52631578947368421(2 X 2 =4,+1=5, 5 put to left)

18.       ₁052631578947368421(5 X 2 =10, 1 carried over, 0 put to left)

Now from step 18 onwards the same numbers and order towards left continue.

Thus 1 / 19 = 0.052631578947368421

Example: Value of 1 / 7.

1/7 में हर के अंक को 9 बनाने के लिए हर और अंश में 7 से गुणा करते हैं।

(1/7 = 7/49); हर के 49 का पूर्वेण हैं, 4; जिसका एकाधिक 5 हैं। भिन्न के आवृति दशमलव स्वरूप का अन्तिम अंक 7 होगा तथा 7-1 = 6 अंकों के पश्चात् दशमलव अंकों की पुनरावृति होगी।

1.      7

2.      ₃57 (multiply 7 by 5 =35, 3 carried over, 5 put to left)

3.      ₂857 (5 X 5 =25, +3 = 28, 2 carried over, 8 put to left)

4.      ₄2857 (8 X 5 =40, +2 = 42, 4 carried over, 2 put to left)

5.      ₁42857 (2 X 5 =10, +4 = 14, 1 carried over, 4 put to left)

6.      ₂142857 (4 X 5 =20, +1 = 21, 2 carried over, 1 put to left)

अतः

Vedic Mathematical Formula

Vedic Mathematical Formula

(वैदिक गणित के सूत्र)

Sutras

1.    Ekadhikena Purvena (एकाधिकेन पूर्वेण)

पहले से एक अधिक के द्वारा।

‘By one more than the previous one’

Corollary: Anurupyena

2.    Nikhilam navatascaramam Dasatah (निखिलम् नवतश्चरमं दशतः)

सभी नौ में से परन्तु अन्तिम दस में से।

‘All from nine and last from ten’

Corollary: Sisyate Sesasamjnah

3.    Urdhva – tiryagbhyam (ऊर्ध्व तिर्यग्भ्याम्)

सीधे और तिरछे दोनों प्रकार से।

‘Vertically and crosswise’

Corollary: Adyamadyenantyamantyena

4.    Paravartya Yojayet (परावर्त्य योजयेत्)

पक्षान्तरण कर उपयोग में लेना।

‘Transpose and apply’

Corollary: Kevalaih Saptakam Gunyat

5.    Sunyam SamyaSamuccaye (शून्यं साम्य समुच्चये)

समुच्चय समान होने पर शून्य होता हैं।

‘When the samuchayas are same, then it is Zero’

Corollary: Vestanam

6.    Anurupye – Shunyamanyat (आनुरूप्ये शून्यमन्यत्)

अनुरूपता होने पर दूसरा शून्य होता हैं।

‘If one is in ratio, the other one is zero’

Corollary: Vestanam

7.    Sankalana – Vyavakalanabhyam  (संकलन-व्यवकलनाभ्याम्)

जोड़कर और घटाकर।

‘By addition and subtraction’

Corollary: Yavadunam Tavadunikritya Vargancha Yojayet

8.    Puranapuranabhyam (पूरणापूरणाभ्याम्)

अपूर्ण को पूर्ण करके।

‘By completing’

Corollary: Antyayordashake'pi

9.    Chalana - Kalanabhyam (चलन-कलनाभ्याम्)

चलन-कलन के द्वारा।

‘By calculus’

Corollary: Antyayoreva

10.  Yavadunam (यावदूनम्)

जितना कम हो, अर्थात् विचलन।

‘By the deficiency’

Corollary: Samuccayagunitah

11.  Vyastisamastih (व्यष्टिसमष्टिः)

एक को पूर्ण तथा पूर्ण को एक मानते हुए।

‘Whole as one and one as whole’

Corollary: Lopanasthapanabhyam

12.  Sesanyankena charamena (शेषाण्यड्केन चरमेण)

अंतिम अंक से अवशेष को।

‘Reminder by the last digit’

Corollary: Vilokanam

13.  Sopantyadvayamantyam (सोपान्त्यद्वमन्त्यम्)

अन्तिम और उपान्तिम का दुगुना।

‘Ultimate and twice the penultimate’

Corollary: Gunitasamuccayah Samuccayagunitah

14.  Ekanyunena Purvena (एकन्यूनेन पूर्वेण)

पहले से एक कम के द्वारा।

‘By one less than the previous one’

Corollary: Dhwajanka

15.  Gunitasamuchayah (गुणितसमुच्चयः)

गुणितों का समुच्चय।

‘The whole product (The product of the sums)’

Corollary: Dwandwa Yoga

16.  Gunakasamuchayah (गुणकसमुच्चयः)

गुणकों का समुच्चय।

‘Set of multipliers (All the multipliers)’

Corollary: Adyam Antyam Madhyam

Sub-Sutras

1.    Anurupyena (आनुरूप्येण)

अनुपातों से।

‘Proportionality’

2.    Sisyate-Sesasmjnah (शिष्यते शेषसंज्ञः)

एक विशिष्ट अनुपात में भाजक के बढ़ने पर भजनफल उसी अनुपात में कम होता हैं तथा शेषफल अपरिवर्तित रहता हैं।

‘Quotient decreases in same ratio as divisor increases and remainder remain constant’

3.    Adyamadyen –Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)

प्रथम को प्रथम के द्वारा तथा अन्तिम को अन्तिम के द्वारा।

‘The first by the first and the last by the last’

4.    Kevalaih saptakam-Gunyat (केवलैः सप्तकं गुण्यात्)

7 के लिए गुणक 143

‘For 7 the Multiplicand is 143’

5.    Vestanam (वेष्टनम्)

आश्लेषण करके।

‘By ousculation’

6.    Yavadunam Tavadunam (यावदूनम् तावदूनम्)

विचलन घटा करके।

‘Subtract by the deficiency’

7.    Yavadunam Tavadunikrtya Varganca Yojayet (यावदूनम् तावदूनीकृत्य वर्ग च योजयेत्)

संख्या की आधार से जितनी भी न्यूनता हो उतनी न्यूनता और करके उसी न्यूनता का वर्ग भी रखें।

‘What ever the deficiency subtract that deficit from the number and write along side the square of that deficit’

8.    Antyayor Dasake’pi (अन्त्ययोर्दशकेऽपि)

अन्तिम अंकों का योग 10 वाली संख्याओं के लिए।

‘Numbers of which the last digits added up give 10’

9.    Antyayoreva (अन्त्ययोरेव)

अन्तिम पद से ही।

‘Only the last terms’

10.  Samuchayagunitah (समुच्चयगुणितः)

गुणनफल की गुणन संख्याओं का योग।

‘The sum of the products’

11.  LopanaSthapanabhyam (लोपनस्थापनाभयाम्)

विलोपन तथा स्थापना से।

‘By alternate elimination and retention’

12.  Vilokanam (विलोकनम्)

देखकर।

‘By mere observation’

13.  Gunita Samuccayah Samuccaya Gunitah (गुणितसमुच्चयः समुच्चयगुणितः)

गुणनखण्ड़ो की गुणन संख्याओं के योग का गुणनफल गुणनफल की गुणन संख्याओं के योग के समान होता हैं।

‘The product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product (The product of the sum is the sum of the products)’

14.  Dhwajank (ध्वजांक्):

ध्वज लगाकर।

‘On the flag’