Brunton39

2022-06-25

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

Trey Ross

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)
$180=2\cdot 90={2}^{2}\cdot 45={2}^{2}\cdot 3\cdot 15={2}^{2}\cdot {3}^{2}\cdot 5\phantom{\rule{0ex}{0ex}}100=2\cdot 50={2}^{2}\cdot 25={2}^{2}\cdot {5}^{2}\phantom{\rule{0ex}{0ex}}\frac{{2}^{2}\cdot {3}^{2}\cdot 5}{{2}^{2}\cdot {5}^{2}}=\frac{{3}^{2}}{5}=\frac{9}{5}$

Fletcher Hays

By the Euclidean algorithm $\left(180,100\right)=\left(80,100\right)=\left(80,20\right)=\left(0,20\right)=20.\phantom{\rule{thinmathspace}{0ex}}$. Cancel this gcd.
Euclid's algorithm is very efficient, so generally this method will be quickest.

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