So I want to try and divide numbers that have decimals in the answers. Such as 27/2. If I tried to do this mentally I would get 13 Remainder of 1 but how do I figure out that it equals 13.5 exactly all in my head (Of course 27/2 is quite simple but maybe 153/13)
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For decimal results you just keep dividing beyond the decimal point.
27.0/2 = 10.0 + 7.0/2 = 10.0 + 3.0 + 1.0/2 = 10.0 + 3.0 + 0.5 =13.5
Let’s just do this.
I assume you see that this goes into 13 at least 10 times
Subtract 10 times 13 or 130 from 153 to get 23.
We can subtract another 13 from 23 to get 10.
We are now at the answer:
11 r 10.
For every next digit after the decimal point we add a zero to the remainder.
So 10 becomes 100. This goes into 13 7 times. 7 times 13 is 91. Remainder is 9.
9 becomes 90. We just calculated 91 (7x13).
Since 91 is almost 90, we can round to 7 times and be done with it.
Answer in this case is 21.77
If we need lots of digits we can continue. There is also a shortcut, but I’ll keep that for later.
Let’s continue first.
So remainder is 90. 90 goes into 13 6 times.
90-78=12 => 120.
120 goes into 13 9 times. 120-117= 3. 3 becomes 30. 2 times. 26. 4 becomes 40 etc.
3 times make 39. Remainder is 1.
We can write a digit of 0. 1 becomes 10 and we are back where we started.
Rinse, repeat. Now we have unlimited digits.
Answer is: 11.76923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923076923769230769230…
Now for the shortcut.
Read back from the line that says “Since 91 is almost 90…”.
I wrote about working with negative remainders a long time ago. Do a search.
Remainder is 90. 13 times 7 makes 91, remainder is -1.
-1 becomes -10 (digit is 0) and -10 is our (negative) starting point for the first digit after the decimal point.
Focus on the 77.
For subtraction, think of 77 as 76.9999999… which is the same. Really, it is.
Now I hope you see the subtraction is easy:
11.769999999999…
00.007699999999… (subtract 76999 3 decimal places further).
___________________ -
11.7692300000000…
Mentally do:
7699999....
___76999....
------------------
76923000....
In this procedure we started with a remainder of one and ended with minus one.
Since we are subtracting a remainder of one - which ends in minus one - subtracting [minus one] is the same as adding one.
So without the decimal point we get:
1176923 R 1
R1 means we get the decimal expansion of 076923 076923 076923 076923 …
The shortcut can quickly generate the ‘769230’.
Maybe it helps to see 76923 X 13 = 999,999 or 1 million R1.
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If my previous post is unclear, in the past I wrote a lot about division and also about negative remainders:
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