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In this paper, we consider the sum‐product problem of obtaining lower bounds for the size of the setA  +  A    A  +  A    :  =a  +  b    c  +  d    :  a  ,  b  ,  c  ,  d  ∈  A  ,  c  +  d  ≠  0   ,for an arbitrary finite set   A   of real numbers. The main result is the boundA  +  A    A  +  A≫|  A  |    2  +  2  /  25|  A  :  A  |    1  /  25    log   |  A  |,where    A  :  A    denotes the ratio set of   A  . This improves on a result of Balog and the author (‘New sum‐product estimates for real and complex numbers’,  Discrete Comput. Geom . 53 (2015) 825–846), provided that the size of the ratio set is subquadratic in    |  A  |   ; that is, we establish that (inline equation)A  +  A    A  +  A≪    |  A  |   2   ⇒   |  A  :  A  |   ≫|  A  |   2log  25    |  A  |.This extremal result answers a question similar to some conjectures in a recent paper of the author and Zhelezov (‘A bound on the multiplicative energy of a sum set and extremal sum‐product problems’,  Moscow J. Comb. Number Theory  5 (2015) 53–70).

If ( A + A )/( A + A ) is small, then the ratio set is large