The orbital parameters of binaries at intermediate periods (10(2)-10(3) d) are difficult to measure with conventional methods and are very incomplete. We have undertaken a new survey, applying our pulsation timing method to Kepler light curves of 2224 main-sequence A/F stars and found 341 non-eclipsing binaries. We calculate the orbital parameters for 317 PB1 systems (single-pulsator binaries) and 24 PB2s (double-pulsators), tripling the number of intermediate-mass binaries with full orbital solutions. The method reaches down to small mass ratios q approximate to 0.02 and yields a highly homogeneous sample. We parametrize the mass-ratio distribution using both inversion and Markov-Chain Monte Carlo forward-modelling techniques, and find it to be skewed towards low-mass companions, peaking at q approximate to 0.2. While solar-type primaries exhibit a brown dwarf desert across short and intermediate periods, we find a small but statistically significant (2.6 sigma) population of extreme-mass-ratio companions (q < 0.1) to our intermediate-mass primaries. Across periods of 100-1500 d and at q > 0.1, we measure the binary fraction of current A/F primaries to be 15.4 per cent +/- 1.4 per cent, though we find that a large fraction of the companions (21 per cent +/- 6 per cent) are white dwarfs in post-mass-transfer systems with primaries that are now blue stragglers, some of which are the progenitors of Type Ia supernovae, barium stars, symbiotics, and related phenomena. Excluding these white dwarfs, we determine the binary fraction of original A/F primaries to be 13.9 per cent +/- 2.1 per cent over the same parameter space. Combining our measurements with those in the literature, we find the binary fraction across these periods is a constant 5 per cent for primaries M-1 < 0.8 M-circle dot, but then increases linearly with log M-1, demonstrating that natal discs around more massive protostars M-1 greater than or similar to M-1(circle dot) become increasingly more prone to fragmentation. Finally, we find the eccentricity distribution of the main-sequence pairs to be much less eccentric than the thermal distribution.