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.