How do you evaluate 114\div 268114÷268?

1 Answer
Feb 10, 2017

114/268 = 0.4bar(253731343283582089552238805970149)114268=0.4¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯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)2681:0268
2: color(white)(0)5362:0536
3: color(white)(0)8043:0804
4: 10724:1072
5: 13405:1340
6: 16086:1608
7: 18767:1876
8: 21448:2144
9: 24129:2412

Then we find:

enter image source here

In this example, note that the remainder 6868 eventually repeats, so the digits of the quotient will repeat from that point onwards too.

So we find:

114/268 = 0.4bar(253731343283582089552238805970149)114268=0.4¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯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 00's).