The Padé-Laplace technique consists of approximating impulse response relations by fitting the Laplace transforms of such relations as ratios of polynomials in the transform variable. This can be used to define "reduced models" that capture the dominant behaviour of more complex systems. This approach is illustrated by analysing various aspects of the carbon cycle and its connection to climate, providing a way to capture how the interactions depend on the timescales involved. The Padé-Laplace technique is used to relate descriptions of the carbon cycle in terms of impulse response functions versus descriptions in terms of feedbacks. It is also used to discuss the concept of CO ${}_2$-emission equivalence. A further example analyses the gain of the climate-carbon feedback loop. This is approximated with a simple parameterization that captures the results of more complex model results and shows that the gain on timescales of centuries is as much as 3 times the gain on decadal timescales. The scope for extensions to more general aspects of the carbon system, such as the distribution of radiocarbon, is noted along with other potential extensions of this approach.