VEDIC MATHS-87

 

VEDIC MATHS

                           By OMKAR TENDOLKAR

Hello friends,

                      This is  post number 87 from the series of "Vedic maths" blogs. Here in this blog we will learn about "Auxilliary Fraction-IV"

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".

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 1:

Now, let us consider the fractions where the divisors are numbers ending with 1.

E.g. 21, 31, 51, 71 etc.

General Steps:

• Subtract 1 from the numerator
• Subtract 1 from 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 by using the new divisor
• Every division should return a quotient and a remainder and should not be a decimal division
• Subtract the quotient from 9 at each step to form the next dividend.

Example :

1.Convert the fraction 4 / 21 to its decimal form.

Let us now see the details for 4 / 21

Steps:

• Subtract 1 from the numerator to get 3
• Subtract 1 from the divisor to get 20
• The modified division now is 3 / 20
• Remove the zero from the denominator by placing a decimal point in the numerator
• The modified division is 0.3 / 2
• We will not write the final answer as 0.15 but We will carry out a step-by-step division as explained below

Steps for step-by-step division

Step 1:

4 / 21 = 3 / 20 = 0.3 / 2.

Step 2:

Here the starting point is 0.3 ÷ 2. As we can divide 3 by 2, we give a decimal point in the answer with 1 as quotient and remainder 1 to be written. This means our next dividend is .       

4 / 21 = 0. 1......

               1

Step 3:

Subtract the quotient (1) from 9 and write it down Now, the next number for division is 18 and not 11.

                      8  

                   .  

                 .

4 / 21 = 0. 1......

                1

On dividing 18 by 2, we get the q = 9 and r = 0. The result now looks as

                      8  

                   .  

                 .

4 / 21 = 0. 1 9......

               1 0

Step 4:

Subtract the quotient (9) from 9 and write it down Now, the next number for division is 0 and not 9.

                      8  0  9 

                   .   .   .

                 .   .   .

4 / 21 = 0. 1 9 0......

               1 0 9

Step 5:

The final result upto 5 digits of accuracy

                      8  0  9  5  2

                   .   .   .   .  .

                 .   .   .  .  .

4 / 21 = 0. 1 9 0 4 7......

               1 0 0 1 1

4/21 = 0.1 9 0 4 7

Our answer becomes 0.19047.

Answer:

4 / 21 = 0.19047

.

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 / 111 to its decimal form.

7 / 111 = 6 / 110 = 0.6 / 11

                     9  3  6  9  3

                   .   .   .   .  .

                 .   .   .  .  .

7 / 111 = 0. 0 6 3 0 6......

               6 3 0 6 3 

Our answer becomes 0.063063.

Answer:

7 / 111 = 0.063063.

Solve the following example.

1. 13 / 31

Ans : 13 / 31 = 12 / 30 = 1.2 / 3 = 0.419355.

2. 3 / 31

Ans : 3 / 31 = 2 / 30 = 0.2 / 3 = 0.09677.

3. 2 / 51

Ans :  2 / 51 = 1 / 50 = 0.1 / 5 = 0.03921.

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

You may try following examples.

1. 4/61

2. 5/71

3. 8/41

4. 7/11

5. 5/81

In next blog we will discuss about "Auxilliary Fraction-V".     

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.

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