VEDIC MATHS-89

 

VEDIC MATHS

                           By OMKAR TENDOLKAR

Hello friends,

                      This is  post number 89 from the series of "Vedic maths" blogs. Here in this blog we will learn about "Auxilliary Fraction-VI"

Reference:

We had already learn about "Types Of Decimals" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-83".

We had already learn about Reciprocals of Numbers Ending in 9 in "Auxilliary Fraction-I" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-84".

We had already learn about Reciprocals of Numbers Ending in 3 in "Auxilliary Fraction-II" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-85".

We had already learn about Reciprocals of Numbers Ending in 7 in "Auxilliary Fraction-III" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-86".

We had already learn about Reciprocals of Numbers Ending in 1 in "Auxilliary Fraction-IV" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-87".

We had already learn about Reciprocals of Numbers Ending in 8 in "Auxilliary Fraction-IV" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-88".

Decimals:
Here in this post, we shall look at methods to convert fractions to decimals.

Vedic One-Line Method:

The conventional practice of converting a reciprocal or fraction into a decimal number involves the division of the numerator by the actual denominator. When the denominator is an odd prime number such as 19, 23, 29, 17 etc., the actual division process becomes too difficult; whereas, if we use the Vedic method, the whole division process becomes oral and the decimals of such fractions can be written directly.

We will now see additional techniques which can speed up the division of fractional numbers where :
  • the dividend is less than the divisor and
  • the divisor is small e.g. having 2 or 3 digits and
  • the last digit of the divisor ends with 1, 6, 7, 8 and 9 E.g. of such divisors are 19, 29, 27, 59, 41, 67, 121 etc.
Other divisors ending in 2, 3, 4, 5 and 7 can be converted to such divisors by multiplying by a suitable number.

We will see examples of divisors ending in each of the digits from 1 to 9 to get a good insight into the entire process.

Reciprocals of Numbers Ending in 6:

Let us consider cases where the divisor ends with 6.
E.g. 26, 36, 56, 86 etc.

General Steps:
  • Add 4 to the denominator and use that as the divisor
  • Remove the zero from the divisor and place a decimal point in the numerator at the appropriate position
  • Carry out a step-by-step division as explained before
  • Every division should return a quotient and a remainder and should not be a decimal division
  • The quotient and the remainder is taken as the next base dividend as explained before
  • Since the last digit of the divisor is 6 which has a difference of 3 from 9, at each stage, we multiply the quotient by 3 and add to the base dividend to compute the next gross dividend.
Example :
1.Convert the fraction 6 / 76 to its decimal form.

Steps:
  • Add 4 to the divisor to get 80
  • The modified division is now 6 / 80
  • Remove the zero from the denominator by placing a decimal point in the numerator giving the modified division as 0.6 / 8
  • We will not write the final answer as 0.075 but
  • We will now carry out a step-by-step division as explained below
Steps for step-by-step division
Step 1:
6 / 76 = 6 / 80 = 0.6 / 8.

Step 2:
Here the starting point is 0.6 ÷ 8. As we can not divide 6 by 8, we give a decimal point in the answer with 0 as quotient and remainder 6 to be written. This means our next dividend is .       

6 / 76  = 0. 0......
                6

Step 3:
Now, the initial base dividend is 60 Multiply the quotient digit (0) by 3 and add to the base dividend 60 to get 60. This is a redundant step here since the quotient digit is zero, but it is shown for uniformity. The next number for division is 60 which on division by 8 gives q = 7 and r = 4. The result now looks as

6 / 76  = 0. 0 7......
                6 4

Step 4:
Now, the base dividend is 47 Multiply the quotient digit (7) by 3 to get 21 and add it to the base dividend (47) to get 68 The next number for division is 68 which on division by 8 gives q = 8 and r = 4. The result now looks as

6 / 76  = 0. 0 7 8......
                6 4 4

Step 5:
The final result upto 5 digits of accuracy
 6 / 76  = 0. 0 7 8 9......
                6 4 4 0
Our answer becomes 0.07894.

Answer:
6 / 76 = 0.07894
.
NOTE: if this process of division is continued, the same set of digits will start repeating. We stop dividing once we get the required number of decimal places.

2. Convert the fraction 7 / 26 to its decimal form.

7 / 26 = 7 / 30 = 0.7 / 3

7 / 26  = 0. 2 6 9 2......
            
Our answer becomes 0.2692.

Answer:
7 / 26 = 0.2692.

Solve the following example.
1. 1 / 46
Ans : 1 / 46 = 1 / 50 = 0.1 / 5 = 0.02173.

2. 5 / 36
Ans : 5 / 36 = 5 / 40 = 0.5 / 4 0.01388.

3. 5 / 56
Ans :  5 / 56 = 5 / 60 = 0.5 / 6 = 0.089287.

The simplicity of this method can be vouched from examples given above.

You may try following examples.
1. 7/66
2. 5/76
3. 1/46
4. 7/16
5. 3/46

In next blog we will discuss about "Percentages".     

Are you excited for this?...
Then, please wait for it.
I will post my new blog in next week.

We will meet very soon through our next  blog. Till that stay connected, stay healthy and stay safe.

Thanks

for giving your valuable time.

Good day😊

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