Question

When writing 3.7272727272... as a ratio of integers, what is the common ratio?

Answer 1

3.727272... is equal to 3 + 0.727272...

First, we need to find the common ratio for 0.727272..
For an instance, a = 0.727272...
There are 2 repetitive numbers, seven and two. Multiply the decimal fraction by 100.
(Note: if there is only 1 repetitive number, multiply by 10. if there are 3 repetitive numbers, multiply by 1000 and so on)

100a = 72.727272...
a = 0.727272...
-------------------------- - (substract)
99a = 72
a =
`(72)/(99)`
a =
`(8)/(11)`
The common ratio for 0.727272.. is
`(8)/(11)`

Second, find the common ratio for 3.727272...
3.727272...
= 3 + 0.727272...
= 3 +
`(8)/(11)`
=
`(33)/(11) + (8)/(11)`
=
`(41)/(11)`
The common ratio for 3.727272.. is
`(41)/(11)`