Question

How do you divide `\frac { - 5} { 11}` using long division?

Answer 1

`(-5)/11=-0.45bar(45)`
`color(white)("XXX")`see below for solution using long division

Explanation:

This is simpler to see if we do the long division with positive numbers and then negate the quotient. (It is possible to do it with a negative dividend but the process tends to be ore error prone).

Set-up:
`color(white)("XXX")underline(color(white)("XXXXXXXXXX"))`
`11 color(white)("X")) color(white)("X")5.0 color(white)("X")0 color(white)("X")0color(white)("X") 0 color(white)("X")...`

`11` "goes into" `5`, `0` times
Subtract `0xx11` from the dividend
and bring down the next digit of the dividend (`0`)
`color(white)("XXX")underline(color(white)("XXXXXXXXX"))`
`11 color(white)("X")) color(white)("X")5.0 color(white)("X")0 color(white)("X")0color(white)("X") 0 color(white)("X")...`
`color(white)("XXXX")underline(0)`
`color(white)("XXXX")5color(white)(".")0`

`11` "goes into `50`, `4` times
Subtract `11xx4`
and "bring down" the next `0`
`color(white)("XXX")underline(color(white)("X")0.4 color(white)("XXXXXXX"))`
`11 color(white)("X")) color(white)("X")5.0 color(white)("X")0 color(white)("X")0color(white)("X") 0 color(white)("X")...`
`color(white)("XXXX")underline(0)`
`color(white)("XXXX")5color(white)(".")0`
`color(white)("XXXX")underline(4color(white)(".")4)`
`color(white)("XXXX4.")6color(white)("X")0`

Repeat this process:
`color(white)("XXX")underline(color(white)("X")0.4 color(white)("X")5color(white)("XXXXX))`
`11 color(white)("X")) color(white)("X")5.0 color(white)("X")0 color(white)("X")0color(white)("X") 0 color(white)("X")...`
`color(white)("XXXX")underline(0)`
`color(white)("XXXX")5color(white)(".")0`
`color(white)("XXXX")underline(4color(white)(".")4)`
`color(white)("XXXX4.")6color(white)("X")0`
`color(white)("XXXX4.")underline(5color(white)("X")5)`
`color(white)("XXXX4.5"X")5`

Notice that we now have a remainder of `5` just as we did after the first subtraction.
`rarr` the quotient pattern will repeat from this point on...
`color(white)("XXX")underline(color(white)("X")0.4 color(white)("X")5color(white)("X") bar(4 color(white)("X")5))`
`11 color(white)("X")) color(white)("X")5.0 color(white)("X")0 color(white)("X")0color(white)("X") 0 color(white)("X")...`

As noted at the beginning
if `5/11=0.45bar(45)`
then `(-5)/11=-0.45bar(45)`