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Wednesday, June 23, 2010

Vedic Mathematics Lesson 54: Auxiliary Fractions 3

In the previous two lessons, we saw how certain fractions can be converted to what are known as auxiliary fractions that enable us to find decimal representations of them much more easily than by doing long division using the given denominator as the divisor.

You can find all my previous posts about Vedic Mathematics below:

Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
10's Complements
Multiplication Part 1
Multiplication Part 2
Multiplication Part 3
Multiplication Part 4
Multiplication Part 5
Multiplication Special Case 1
Multiplication Special Case 2
Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method
Digital Roots
Straight Division I
Straight Division II
Divisibility Rules
Simple Osculation
Multiplex Osculation
Solving Equations 1
Solving Equations 2
Solving Equations 3
Solving Equations 4
Mergers 1
Mergers 2
Mergers 3
Multiple Mergers
Complex Mergers
Simultaneous Equations 1
Simultaneous Equations 2
Quadratic Equations 1
Quadratic Equations 2
Quadratic Equations 3
Quadratic Equations 4
Cubic Equations
Quartic Equations
Polynomial Division 1
Polynomial Division 2
Polynomial Division 3
Square Roots 1
Square Roots 2
Square Roots 3
Recurring Decimals 1
Recurring Decimals 2
Recurring Decimals 3
Auxiliary Fractions 1
Auxiliary Fractions 2

However, we also found that only certain denominators lend themselves readily to being converted to auxiliary fractional form for using our shortcuts readily. These are denominators that end in a series of 9's or ones that end in a series of zeroes followed by a 1. What can you do if your denominator does not conform to these patterns?

In many cases, if the denominator is not a number that conforms to the patterns above, we may not be able to do anything about it. But in some cases, we may actually be able to convert the fraction into an equivalent one that has a denominator that conforms to the patterns we want.

This is going to be a pretty short lesson, primarily dealing with common multiples of numbers and factors of numbers that conform to the patterns we need for using auxiliary fractions to do division.

Many two-digit numbers ending in 3 (such as 13, 23, etc.) can be converted to a number that ends in 9 simply by multiplying by 3. Thus, 4/13 may be harder to tackle than the equivalent 12/39 (whose auxiliary fractional form is 1.2/4). Similarly 31/43 can be converted to 93/129 (whose auxiliary fractional form is 9.3/13).

Many two digit numbers that end in 3 can also be multiplied by 7 to get a number that ends in 1. Thus, 4/13 can be expressed as 28/91 (which then leads to an auxiliary fraction of 2.7/9). Similarly 21/43 can be expressed as 147/301, whose auxiliary fractional form is 1.46/3.

However, multiplication by large numbers like 7 (which by itself can be somewhat challenging) can result in unwieldy denominators whose auxiliary fractional forms are only slightly easier to work with. For instance, 6/53 can be converted to 42/371, whose auxiliary fractional form is 4.1/37. It is hard to argue that this is an enormous improvement over 6/53! That is also one of the reasons I don't recommend these methods for denominators with more than 2 digits.

By turning the logic of the previous paragraphs the other way around, you can also conclude that for numbers ending in 7, you might be able to get good equivalents by multiplying the numerator and denominator by 3. Thus 23/27 becomes 69/81, whose auxiliary fractional form is 6.8/8. Similarly, 13/67 becomes 39/201, wose auxiliary fractional form is 0.38/2.

Similarly, two-digit numbers ending in 7 can also be multiplied by 7 to get a number that ends in 9. This can lead to good auxiliary fractional forms. For example, 3/7 is 21/49, whose auxiliary fractional form is 2.1/5. Similarly, 32/57 is the same as 224/399, but 224/399 is much easier to work with than 32/57 because its auxiliary fractional form is 2.24/4.

However, there are several cases where it seems impossible to convert a denominator into anything useful from an auxiliary fractional standpoint. For instance, most numbers that end in even numbers or in 5 or an even number can be difficult to convert to an equivalent form that readily yields a simple auxiliary fraction that we can work with. Some other numbers just seem too large to convert into an equivalent form that has a convenient auxiliary fractional form to work with.

But by keeping some handy multiples and factors in mind, it is possible to deal with certain numbers that might otherwise seem formidable.

For instance, 143 seems a pretty unwieldy denominator to work with. And 439 (143*3) is not significantly better in terms of giving us a much better denominator even in auxiliary fractional form. But if you knew that 143*7 = 1001, then you know that it is very easy to work with any number whose denominator is 143: all you have to do is multiply both numerator and denominator by 7 to get a number whose auxiliary fractional form is probably trivial to work with!

Once you know that 143 has this interesting property, it is easy to extend it to most multiples of 143. So, we can say that 286*3.5 = 1001. Similarly, 572*1.75 = 1001. You can even deal with 715, which is a number that ends in 5 by taking advantage of the fact that 715 = 143*5. Thus 324/715 would simply become 64.8/143, which we would then convert to 453.6/1001, giving us an auxiliary fractional form of 0.4535/1.

By the same token, any number that ends in a series of 3's can be converted to a number that ends in a series of 9's by multiplying by 3. Thus 133*3 = 399, 15333*3 = 45999, and so on. Moreover, any number that ends in a series of 6's followed by a 7 can be multiplied by 3 to get a number that ends in a series of zeroes followed by a 1. Try it out: 667*3 = 2001. 26667*3 = 80001. 1666667*3 = 5000001.

A number that ends in a series of 6's can be converted to a number that ends in a series of 3's simply by dividing in half. Thus, 23/1666 can be converted to 11.5/833, which can then be multiplied by 3 to get 34.5/2499. This gives us an auxiliary fraction of 0.345/25. Similarly, you can manipulate 345/2666 into an auxiliary fractional form of 0.5175/4, by dividing by 2 and multiplying by 3.

Many other numbers ending in even numbers can also be dealt with by dividing and multiplying by judicious combinations of numbers to get numbers that conform to the patterns required for computation using auxiliary fractions. If a number ends in a series of zeroes followed by 2, for instance, dividing the numerator and denominator by 2 could be an easy way to convert the denominator into a number with a series of zeroes followed by 1. Thus 233/3002 becomes 116.5/1501, which then yields the auxiliary fractional form 1.164/15.

Hopefully, this lesson has illustrated how auxiliary fractions have wider applicability than the strict rules on the format of their denominators would indicate. You have to be creative about how you multiply and divide by various numbers so that the denominator becomes easier to deal with. Knowing some common multiples and factors of numbers helps in this effort. Being able to do quick multiplications and divisions by single digit numbers will also help in this regard as well as have wide applicability in various walks of daily life involving the use of simple arithmetic. In the next lesson, we will see how the rules regarding the format of denominators can be relaxed to derive auxiliary fractions even for some numbers that don't conform to the rules we have seen so far. In the meantime, good luck, and happy computing!

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