Problem:
Suppose that p and q are positive numbers for which
log9​(p)=log12​(q)=log16​(p+q)
What is the value of q/p?
Answer Choices:
A. 34​
B. 21​(1+3​)
C. 58​
D. 21​(1+5​)
E. 916​
Solution:
Let t be the common value of log9​(p),log12​(q) and log16​(p+q). Then
p=9t,q=12t, and 16t=p+q=9t+12t
Divide the last equation by 9t and note that
9t16t​=(3t4t​)2=(9t12t​)2=(pq​)2
Now let x stand for the unknown ratio q/p. From the division referred to above we obtain x2=1+x, which leads easily to x=21​(1+5​) since x must be the positive root.