Although noticing the repeated pattern of a multiple of 9 in the fraction 0.16327272727272726 naturally suggests multiplying by 11, and then we get the much simpler value 1.796, at which point it's much easier to continue. I wouldn't have broken out a general analysis method for this, although it's neat to know that they exist.
alyxyaJan 26, 2026, 12:14 AM
I think the standard way to convert repeating decimals or decimals that appear to have a certain repeating pattern to fractions is to take the first repeating period and divide by 0.999.. with the number of 9s matching the period length.
0.163272727.. = 0.163+0.00027/0.99 = 163/1000+27/99000 = 449/2750
Although noticing the repeated pattern of a multiple of 9 in the fraction 0.16327272727272726 naturally suggests multiplying by 11, and then we get the much simpler value 1.796, at which point it's much easier to continue. I wouldn't have broken out a general analysis method for this, although it's neat to know that they exist.