Using a factor tree to reduce a fraction? Good Idea?

Question

I am trying to figure out how one reduces 180/100 to 9/5

My factor tree for 180 is 90 *2 - 30*3 -5*6 - 2*3

Thus my prime numbers are 2*3*5 = 30

Maybe I have totally forgotten how to reduce a fraction like this, however, what method should I use if I can not see it automatically?

Answer 1

You should use method such that a number is divisible by $2$ if it's last digit is even,it's divisible by $3$ and $9$ if their sum of digits is divisible by $3$ or $9$(in that order) $$180=2\cdot90=2^2\cdot45=2^2\cdot3\cdot15=2^2\cdot3^2\cdot5\\100=2\cdot50=2^2\cdot25=2^2\cdot5^2\\\frac{2^2\cdot3^2\cdot5}{2^2\cdot5^2}=\frac{3^2}{5}=\frac{9}{5}$$

Answer 2

By the Euclidean algorithm $(180,100) = (80,100) = (80,20) = (0,20) = 20.\,$ Cancel this gcd.

Euclid's algorithm is very efficient, so generally this method will be quickest.