Ratios of top-quark pair to Z-boson cross sections measured from proton-proton collisions at the LHC centre-of-mass energies of s=13 TeV, 8 TeV, and 7 TeV are presented by the ATLAS Collaboration. Single ratios, at a given s for the two processes and at different s for each process, as well as double ratios of the two processes at different s, are evaluated. The ratios are constructed using previously published ATLAS measurements of the tt¯ and Z-boson production cross sections, corrected to a common phase space where required, and a new analysis of Z → ℓ+ℓ− where ℓ = e, μ at s=13 TeV performed with data collected in 2015 with an integrated luminosity of 3.2 fb−1. Correlations of systematic uncertainties are taken into account when evaluating the uncertainties in the ratios. The correlation model is also used to evaluate the combined cross section of the Z → e+e− and the Z → μ+μ− channels for each s value. The results are compared to calculations performed at next-to-next-to-leading-order accuracy using recent sets of parton distribution functions. The data demonstrate significant power to constrain the gluon distribution function for the Bjorken-x values near 0.1 and the light-quark sea for x < 0.02.[Figure not available: see fulltext.]