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.
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
The repeating pattern has 2 digits, and is a multiple of 9. 9 times 11 is 99, so that multiplication gives a "repeating pattern of a multiple of 99" — equivalently, a multiple of a repeating pattern of 99. And I assume you're familiar with the concept of .99999... equaling 1.
Neat analysis.
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.
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
> naturally suggests multiplying by 11
Is this a named concept that I can learn about?
The repeating pattern has 2 digits, and is a multiple of 9. 9 times 11 is 99, so that multiplication gives a "repeating pattern of a multiple of 99" — equivalently, a multiple of a repeating pattern of 99. And I assume you're familiar with the concept of .99999... equaling 1.
Not sure if this helps but check it out
https://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/jack1....