Question

How do you evaluate `114\div 268`?

Answer 1

`114/268 = 0.4bar(253731343283582089552238805970149)`

Explanation:

To get a completely accurate answer, long divide until the remainder repeats.

With a long division of this kind, it is often helpful to write down the single digit multiples of the divisor first:

`1: color(white)(0)268`
`2: color(white)(0)536`
`3: color(white)(0)804`
`4: 1072`
`5: 1340`
`6: 1608`
`7: 1876`
`8: 2144`
`9: 2412`

Then we find:

In this example, note that the remainder `68` eventually repeats, so the digits of the quotient will repeat from that point onwards too.

So we find:

`114/268 = 0.4bar(253731343283582089552238805970149)`

where the bar indicates the sequence of digits which repeat.

Any division of two whole numbers will eventually repeat since the running remainder is always smaller than the divisor. So any rational number is representable as a decimal expansion which eventually repeats or terminates (i.e. repeats `0`'s).