VEDIC MATHS-84

 

VEDIC MATHS

                           By OMKAR TENDOLKAR

Hello friends,

                      This is  post number 84 from the series of "Vedic maths" blogs. Here in this blog we will learn about "Auxilliary Fraction-I"

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".

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 9:

We will first consider reciprocals of numbers ending in 9 such as 19, 29, 39, 79, etc.

Usually, dividing by 19, 29 or 39 is not too easy, but here, we will apply the maths sutra By "One More than the One Before". This basically means one more than the number before the 9.

There is a 4 before the 9, in the case of 49; one more than this is 5. So 5 is our Ekadhika which is ‘one more’.

Similarly, the Ekadhika for 59 is 6.

Since only 9 is dropped, we replace the decimal of the numerator by shifting the decimal one place to the left. So it will become 0.1. Now the process will be to divide 0.1 by the Ekadhika number.

For example, 

with 19, the Ekadhika is 2. So our starting point will be 0.1 divided by 2, which can be orally done instead of remembering the table of 19 for 1 ÷ 19.

Let’s see the division method that will give us the answer from left to right.

General Steps:

• Add 1 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 by using the new dividend
• Every division should return a quotient and a remainder and should not be a decimal division
• Every quotient digit will be used without any change to compute the next number to be taken as the dividend. In all other cases, the quotient digit would be altered by a pre-defined method which will be explained in each case.

Example :

1.Convert the fraction 1 / 19 to its decimal form.

Step 1:

We find the Ekadhika of 1 / 19. After dropping the 9 of 19, we get one more than 1, which is 2.

So this becomes 

1 / 19 = 1 / 20 = 0.1 / 2

Step 2:

Here the starting point is 0.1 ÷ 2. As we cannot divide 1 by 2, we give a decimal point in the answer with 0 as quotient and remainder 1 to be written. This means our next dividend is 10.

1 / 19 = 0. 0......

               1

Step 3:

We divide 10 by 2 which gives us 5 and remainder zero. The sum looks like this:

1 / 19 = 0. 0 5......

               1 0

Step 4:

The quotient obtained is 05. We divide 05 by 2. This gives us our next quotient digit of 2 and remainder of 1.

We prefix 1 before 2 as shown below.

1 / 19 = 0. 0 5 2......

               1 0 1

Step 5:

We keep on dividing by 2 and writing down the quotients and remainders as it has already been shown. And this is what we get:

1 / 19 = 0. 0 5 2 6 3

               1 0 1 0 0

Our answer becomes 0.05263.

Answer:

1 / 19 = 0.05263 .

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 1 / 29 to its decimal form.

Step 1:

We find the Ekadhika of 1 / 29. After dropping the 9 of 29, we get one more than 2, which is 3.

So this becomes 

1 / 29 = 1 / 30 = 0.1 / 3

Step 2:

Here the starting point is 0.1 ÷ 3. As we cannot divide 1 by 3, we give a decimal point in the answer with 0 as quotient and remainder 1 to be written. This means our next dividend is 10.             

1 / 29 = 0. 0......

                 1

Step 3:

We divide 10 by 3 which gives us 3 and remainder 1. The sum looks like this:

1 / 29 = 0. 0 3......

               1 1

Step 4:

We continue our division process by the Ekadhika which is 3. We now divide 13 by 3, which gives us our next quotient digit 4 and remainder 1.

Our sum now looks like this:

1 / 29 = 0. 0 3 4......

               1 1 1

Step 5:

We keep on dividing by 3 and writing down the quotients and remainders as it has already been shown. And this is what we get:

1 / 29 = 0. 0 3 4 4 8 2.

                1 1 1 2 0 2

Our answer becomes 0.034482.

Answer:

1 / 29 = 0.034482.

Solve the following example.

1. 1 / 49

Ans : 1 / 49 = 0.020408.

2. 1 / 79

Ans : 1 / 49 = 0.012658.

3. 1 / 99

Ans : 1 / 99 = 0.010101.

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

You may try following examples.       

1. 5/39

2. 7/59

3. 4/09

4. 7/69

5. 4/89

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

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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