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.