Question

How is 0.1363636363636363636... written as a fraction in simplest form?


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Answer 1

3/22

Explanation:

There's a very neat trick for this kind of problem.

First, choose a letter to represent the number.

n = 0.1363636...

Observe that there is a 2-digit block that repeats ("36"), but that block does not repeat immediately to the right of the decimal point. Do two muliplications, one to put the decimal point in front of the repeating block, the other to put the decimal point after the block.

10n = 1.363636...

1000n = 136.363636...

Here comes the cool part: subtract the first equation from the second.

990 n = 135 See? The repeating decimal parts, all lined up as they are, subtract to zeros.

Finally, n = 135/990 = 15/110 = 3/22