We motivate a measurement of various ratios of W and Z cross sections at the Large Hadron Collider (LHC) at large values of the boson transverse momentum ( p T ≳ M W,Z ). We study the dependence of predictions for these cross-section ratios on the multiplicity of associated jets, the boson p T and the LHC centre-of-mass energy. We present the flavour decomposition of the initial-state partons and an evaluation of the theoretical uncertainties. We show that the W + /W - ratio is sensitive to the up-quark to down-quark ratio of parton distribution functions (PDFs), while other theoretical uncertainties are negligible, meaning that a precise measurement of the W + /W - ratio at large boson p T values could constrain the PDFs at larger momentum fractions x than the usual inclusive W charge asymmetry. The W ± /Z ratio is insensitive to PDFs and most other theoretical uncertainties, other than possibly electroweak corrections, and a precise measurement will therefore be useful in validating theoretical predictions needed in data-driven methods, such as using W (→ ℓν) + jets events to estimate the Z(→ ν) + jets background in searches for new physics at the LHC. The differential W and Z cross sections themselves, dσ /d p T , have the potential to constrain the gluon distribution, provided that theoretical uncertainties from higher-order QCD and electroweak corrections are brought under control, such as by inclusion of anticipated next-to-next-to-leading order QCD corrections.