A little over 4 years ago, I wrote a post inspired by Alexander Craig Aitken's approach to dividing by numbers ending in 9, called Leapfrog Division. It's a remarkably fun and simple technique, so if you haven't already checked it out for yourself, give it a read.
I wrote 3 more posts in that series. Leapfrog Division II dealt with dividing by numbers ending in 1. My last 2 posts before this year were Leapfrog Division III, which dealt with dividing by numbers ending in 8, and Leapfrog Division IV, which dealt with dividing by numbers ending in 2.
These were fine, but the latter 3 posts employed rules that became increasingly cumbersome, and were tricky to apply quickly and without error. Recently, however, I learned a far simpler approach that makes these later posts much simpler!
Let's start with credit where credit is due. The thanks should go to Saurabh G over at Hubpages. He had a post at Owlcation titled Divide Numbers Easily Using Vedic Mathematics: Fast and Easy Division Techniques. Towards the end of the post, there's a section with the heading, “How Do You Use Vedic Division When the Divisor Is More Than One Digit?”. The approach taught with numbers ending in 9 is demonstrated a little differently, but is mathematically identical to the original Leapfrog Division post.
The next section, “Multi-Digit Divisor Ending in 8 Example,” is what really struck me the most. He teaches an almost identical approach, and casually mentions that the quotient needs to be doubled before adding the remainder to the 10s place. Click on that link, read through the example, and then compare that approach to what I taught in Leapfrog Division III. I do mention a doubling idea, but the rules I taught are far more complex.
Using Saurabh G's vedic math approach, here's how his example (73 ÷ 138) would look written in the style of the Leapfrog Division posts. We start with the idea that 14 won't go into 7.3, so we start with a 0 and decimal point at the beginning of the answer, and work from there:
- 73 ÷ 14 = 5 (remainder 3)
(Double the 5 to get 10, add the 3 in the 10s place to get 40) - 40 ÷ 14 = 2 (remainder 12)
(Double the 2 to get 4, add the 12 in the 10s place to get 124) - 124 ÷ 14 = 8 (remainder 12)
(Double the 8 to get 16, add the 12 in the 10s place to get 136) - 136 ÷ 14 = 9 (remainder 10)
(Double the 9 to get 18, add the 10 in the 10s place to get 118) - 118 ÷ 14 = 8 (remainder 6)
(Double the 8 to get 16, add the 6 in the 10s place to get 76, and continue on from here, if desired....)
If you wrote down the quotients (the number in bold above) as you went, you'd have written down 0.52898 at this point, which is correct as far as we've gone.
Just to round things out, he includes a chart showing how to adapt this approach for numbers ending in 9 all the way down to 1! Think about that: It took me 3 years and 4 posts to cover numbers ending with 4 different digits, and increasingly difficult rules. Someone else comes along, teaches 2 examples in 1 post, and leaves his readers confident they can handle 9 different digits.
I tip my hat to you, Saurabh G! Thank you for sharing this approach, and giving me and my readers a simpler and more effective mental division tool.
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1 Response to Leapfrog Division V: I've Been Schooled!
NO . Your method works . but saurabh's doesn't work. for example could you divide 21 by 22 with saurabh' method ? the answer is no. The checking process which you put into leapfrog IV is great ! it prevents from making mistake. could you please explain how to create such a checking process for numbers end in 4, 5, 6, 7 ?
Could you please explain how you devised the leapfrog division method ? esp. for dividend which ends in 2 ? for example for what reason we first reduce the numerator by 1 and then start the process? for what reason we double the quotient and take the ones and reduce it by 9 ? does this work for other dividends which ends in other numbers such as 4, 5, 6, 7 ?
how should we change the rules for leapfrog IV so that it works for other dividends ?
What about if the dividend is a three digit number like : 124, 328, ... can we use the leapfrog multiple times ? for example if the problem is : 2535 / 328 we do this : 2535 / 33 then make this 2535 / 3 and use leapfrog.
If you would post another article with "leapfrog division VI" and explain every details in it that would be amazing!
also I found a link between your leapfrog method and this video : https://www.youtube.com/watch?v=FEfyqZMvnHM
but again if we make use of checking process in that video, we never make mistake and the procedure becomes easier.also note that the video doesn't work for the decimal part of the number.
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