Leapfrog Division II

Published on Sunday, April 20, 2014 in , , , ,

Melchoir's, AzaToth's, Mets501's, and Sopoforic's 0.999 perspective imageA little over a year ago, I wrote up my Leapfrog Division post. It's about time for a modification!

The original technique showed you how to work out the decimal equivalent for fractions whose denominators end in 9. In today's post, you'll learn a similar technique for denominators ending in 1.

Before learning this technique, you do want to be very comfortable working with the original Leapfrog Division technique, as it has all the basics of this modified version.

In that post, we worked out 1319, so I thought it might keep things simple if we used 1321 for our first example this time around. Just like before, we're going to use 20 as a base to divide, and minimize it to the number 2 (simply by dropping the 0), but there are a few changes.

Before any dividing, the next step this time is to take the numerator and subtract 1. So, we've taken 1321, converted it to 1320, dropped the zero to get 132. Subtracting 1 from the numerator at this point gives us 122 as our starting point.

Do this calculation in your head so that you get a quotient and a remainder, just as in the original technique: 12 ÷ 2 = 6 (remainder 0)

Just as in the original, you're going to have the remainder “leap in front of” the quotient, but here's where the new extra step comes in. Before the “leaping” is done, you're going to subtract the quotient from 9, then put the remainder in front of that new result.

With our 6 (remainder 0) example, we'd work out 9 - 6 (the quotient) = 3, and then put the remainder of 0 in front of that to get 03. You keep repeating the steps in this manner as far as you wish. Starting from the 122 step:

  • 12 ÷ 2 = 6 (remainder 0)
  • 03 ÷ 2 = 1 (remainder 1)
  • 18 ÷ 2 = 9 (remainder 0)
  • 00 ÷ 2 = 0 (remainder 0)
  • 09 ÷ 2 = 4 (remainder 1)
  • 15 ÷ 2 = 7 (remainder 1)
  • 12 ÷ 2 = 6 (remainder 0)
  • 03 ÷ 2 = 1 (remainder 1)
Do you see how the new number to divide by 2 at each point. In each case, the tens digit is the remainder from the previous problem, and the ones/units digit is 9 minus the quotient from the previous problem. This is a little tricky until you get familiar and comfortable with the technique.

At this point, we can see that the pattern is already starting to repeat, which happens often. Checking with Wolfram|Alpha, we confirm that 1321 ≈ 0.61904761...

You would deal with improper fractions just as before, reducing them to mixed fractions (the links about improper fractions in the original post are still helpful here) before using the leapfrog technique. 7131, for example, should be converted to 2931 as the first step. So we have 2 something, and then we use this version of the leapfrog division technique to work out the decimal equivalent.

In this example, 931 becomes 930, which then becomes 93, and subtracting 1 from the numerator gives us 83. From here, we work out:
  • 08 ÷ 3 = 2 (remainder 2)
  • 27 ÷ 3 = 9 (remainder 0)
  • 00 ÷ 3 = 0 (remainder 0)
  • 09 ÷ 3 = 3 (remainder 0)
  • 06 ÷ 3 = 2 (remainder 0)
  • 07 ÷ 3 = 2 (remainder 1)
  • 17 ÷ 3 = 5 (remainder 2)
  • 24 ÷ 3 = 8 (remainder 1)
No, I don't see a pattern repeating yet, but that's more than enough digits to be impressive. Sure enough, Wolfram|Alpha confirms that 7131 ≈ 2.29032258...

If you take the time to become comfortable with the original version, and then this version, you have 2 very powerful tools for converting decimals to fractions. As a matter of fact, they may be more powerful than you think!

BONUS: Long-time Grey Matters readers may remember my posts on estimating square roots of non-perfect squares and a few tips and tricks. This square root estimation results in an answer in fraction form. Because you're always adding effectively adding two consecutive integers to get the denominator, you'll always have a fraction with an odd denominator.

So, the denominator in that feat will always end in either 1, 3, 5, 7, or 9. Thanks to both versions of the leapfrog division technique, you can now convert any denominators ending in 1 or 9. What about the others numbers?

It turns out that for denominators ending in 3, you can simply multiply the numerator and denominator by 3 to get an equivalent fraction with a denominator ending in 9. For example, 513 = 1539, so you can use the original version of the leapfrog division technique to work that out.

Similarly, when you have a denominator ending in 7, you can simply multiply the numerator and denominator by 3, which will give you a denominator ending in 1. 1217 = 3651, which should be easy for you with this second version of the technique.

For fractions with denominators ending in 5, this technique isn't often applicable. You may get lucky and be able to scale a fraction down to a number ending in 1 or 9, such as 2055 = 411, or 35145 = 729, but you can't always count on that. Using any whole number to scale a fraction up whose denominator ends in 5, of course, can only result in a denominator ending in 0 or 5, of course. If the denominator is just 5, though, you should have little problem working out the decimal equivalent.

Since you now realize you can handle odd denominators ending in 1, 3, 7, or 9, for the square root estimation feat, you stand a good 80% chance of being able to give a decimal equivalent, and taking the feat to an impressive new level!

I suggest practicing this, having fun with it, and impressing a few friends with your newfound skill. Enjoy!

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