Mental math: Division of remainders

[sony]

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

 

[tiny-tim]

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!

For 22/45, the trick is easy …

make it 44/90.

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)

 

[HallsofIvy]

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...

 

[sony]

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