Mental math: Division of remainders

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Discussion Overview

The discussion revolves around mental math techniques for performing division, specifically focusing on how to handle remainders in division problems such as 22/45, 8/31, and 22/39. Participants explore methods to derive exact decimal representations from these divisions without relying on calculators.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses a desire to improve mental math skills, particularly in dividing numbers and understanding remainders.
  • Another participant suggests a method of transforming 22/45 into 44/90 as a way to simplify the division process.
  • There is mention of Vedic mathematics as a potential approach for handling similar divisions, though it is unclear how it applies specifically to the examples given.
  • A participant provides a detailed explanation of the long division process, emphasizing the significance of recognizing repeating remainders and how they lead to the decimal representation of 0.48888... for 22/45.
  • One participant acknowledges a newfound understanding of using long division to determine decimal values from remainders, indicating a learning moment from the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a single method for dividing remainders, as various approaches are suggested, and some participants express confusion about the concepts involved. The discussion remains open-ended with multiple viewpoints presented.

Contextual Notes

Some participants may have different levels of familiarity with long division and mental math techniques, which could influence their understanding of the methods discussed. There is also a lack of clarity on how Vedic mathematics specifically applies to the examples provided.

sony
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I'm working on moving away from long time slavery to the calculator

Most forms of division and multiplication is going well now (long division and in some instances lattice multiplication)

One thing I DON'T get though (and can't find any guides for) is dividing remainders.

For example: How do I figure out 22/45? I can see that answer is around 1/2 but HOW can i work it out to exactly 0.488888...?

Some goes for 8/31, 22/39 etc etc...

Thank you
 
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sony said:
For example: How do I figure out 22/45? I can see that answer is around 1/2 but HOW can i work it out to exactly 0.488888...?

Some goes for 8/31, 22/39 etc etc...

Hi sony! :smile:

For 22/45, the trick is easy …

make it 44/90. :smile:

For the others, I think you're going to need Vedic mathematics,

unless you're happy with approximate results, in which case …

8/31 = (8/30)(30/31) ~ (8/30)(29/30)

22/39 = (22/40)(40/39) ~ (22/40)(41/40) :wink:
 
I'm not clear on what, exactly, your question is. You say
How do I figure out 22/45? I can see that answer is around 1/2 but HOW can i work it out to exactly 0.488888...?
So are you saying that you can do long division but want to know how you know the "8" will always repeat?

4 divides into 22 5 times but 5*45= 225> 220 so we "try" 4 as a quotient instead. 4*45= 180 which is less than 220: the quotient begins ".4 ". Then 220- 180= 40 so we next have to divide 40 by 45. A "trial divisor" of 8 gives 8*45= 360: that is slightly less than 400 so we now know the quotient starts ".48 ". 400- 360= 40. Now, here is the critical point: That is exactly the same remainder we got before. We don't have "try" any trial divisors- we already know that 45 will go into 400 8 times with remainder 40. If we were to continue on indefinitely, we would never get anything other than a quotient of 8 with a remainder of 40. That tells you that the exact value is 0.48888...
 
tiny-tim, Thank you, I found a guide on Wikipedia that proved very helpful.

HallsofIvy: Then I obviously didn't know long division, I didn't know you could use that to figure out the decimal value of the remainder. Your explanation helped.

Thanks
 

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