It's pretty easy to see that both numbers are divisible by 5, so
let's start there. You can fit 5 into 30 a grand total of 6 times, and
45 divided by 5 is 9. So our new fraction is . But hold the phone!
(We're speaking figuratively—you can put the phone down now.) Each of
those numbers is divisible by 3. Do our thing one more time and
we're left with , which can no longer be reduced. Not without a
hacksaw, anyway.
Example 2
Reduce to its lowest terms.
We've got a couple of even numbers here, so clearly 2 will divide
both the numerator and denominator. We start by reducing this fraction
to , but we've still got two even numbers, so we do it again. Now
we've got . There are no natural numbers (except 1) that divide 30
and 23, so the fraction is in lowest terms.
to its lowest terms.
. But hold the phone!
(We're speaking figuratively—you can put the phone down now.) Each of
those numbers is divisible by 3. Do our thing one more time and
we're left with 
, which can no longer be reduced. Not without a
hacksaw, anyway.
to its lowest terms.
, but we've still got two even numbers, so we do it again. Now
we've got 
. There are no natural numbers (except 1) that divide 30
and 23, so the fraction 
is in lowest terms.