Why do I get two different answers when doing long division based on decimal or remainder?

Question

So when I finish calculating 5/1853 using long division, my work looks like this.

    370 r3  (ie, "370, with remainder 3")
  ______
5 )1853
  -15
    35
  - 35
   - 03
    - 0
      3

But when I chuck it into a calculator I get 370r6 so I figured it might have something to do with how calculators round up/down certain numbers.

So I went to a website which might be able to show me a more "correct" answer, so I headed over to CalculatorSoup.com's "Long Division Calculator with Remainders" page which gave me 370r3. But then on the same website they offer the same calculator but with decimals , and when using that one it gave me 370r6 no matter if it was 4 decimals or 1.

Did I get it right or is there something that I'm missing and don't understand fully?

Answer 1

In simpler words, the calculator did not show you 370r6. It showed you $370.6$ which you wrongly interpreted as 370r6. There is a difference between the two. Consider: $$\frac{1853}{5}=370.6 = 370+\color{blue}{0.6}$$and $$\frac{1853}{5} = \frac{1850+\color{blue}{3}}{5} = 370+\frac35$$ It is clear that $3$ is the remainder, while $0.6$ is the fractional part (or the part less than one). Also, notice that a remainder of $6$ while dividing by $5$ is impossible.

In general, whenever you use calculators, remember that after a division calculation, the number after the decimal point is not the remainder. However, to find the remainder, multiply the number after the decimal point (along with the point and a zero, that is, $0.6$ and not $6$) with the divisor ($5$ in this case). (After multiplying, if you get another decimal, then round it to the nearest integer).

Answer 2

Division with Integers ($X/Y$) might give Exact Answer , when $X$ (eg $1850$) is a multiple of $Y$ (Eg $5$) , where the Answer (Eg $370$) is easy to calculate with the long Division Method. Otherwise , we will get a remainder.

Your Case ($1853/5$) is the Example with remainder.
The Answer is $370$ with reminder $3$.

Division with Decimals ($X/Y$) will always give Exact Answer with no Concept of reminder.
When $X$ is a multiple of $Y$ , the fractional Part is $0$ , thus we will get $0$ after the Decimal Point , Eg $1850/5=370=370.0=370.00=370.000=370.0000=370.00000=370.000000$.

When $X$ is not a multiple of $Y$ , the fractional Part is not $0$ , thus we will get some Digits after the Decimal Point , Eg $1853/5=370.6=370.60=370.600=370.6000=370.60000=370.600000=370.6000000$.
With Decimals , the reminder Part must also get Divided into fractional Parts.
There is no Concept of reminder here.

It may happen that the Decimal Value has lots of Digits :
Eg $15/8=1.875$ , where $1$ is the Integer Part & $0.875$ is the fractional Part.
Eg $189/96=1.96875$
We have no reminder here.

It may happen that the Decimal Value has unending Digits :
$2/3=0.666666666........$
$159/7=22.428571428571428571428571428571........$
We have no reminder here too.

In your Case :
It is not $370r6$
It is $370r3$ with reminder
It is $370.6$ with Decimals , there is no reminder $6$ , it is the fractional Part.