suppose a   is directly proportional to b, but inversely proportional to c. if  a=2 when b=5  and c=9, then what c is when b=3?

We will have 2 different equations that we will combine into 1, but we need to find the 2 k values first.  If a is directly proportional to b, then a = kb.  When a = 2 and b = 5, then k = 2/5.  Store that for a minute.  Now onto the second equation to find the other k value.  If a is inversely proportional to c, then a=k/c.  If a = 2 when c = 9, then k = 18.  We will now use the transitive property: if a=kb and a = k/c, then kb=k/c.  The k on the left is 2/5 and we are told that b = 3; the k on the right is 18 and we are looking for c.  So here is what we have: and .  Cross multiply to get 6c = 90 and c = 15.

The given relation means
   a = k·b/c
Filling in the given information, you have
   2 = k·5/9
   k = 18/5 = 3.6

Then for a different value of b with the same value of a, we can find c.
   2 = 3.6·3/c
   c = 3.6·3/2
   c = 5.4