The diagram shows two vessels, $$\(\mathrm{P}\)$$ and $$\(\mathrm{Q}\)$$, both with sides inclined at $$\(45^{\circ}\)$$ to the horizontal. Vessel $$\(P\)$$ tapers outwards and vessel $$\(Q\)$$ tapers inwards, as shown. Both vessels contain a liquid. The depth of the liquid in the vessels is the same. The liquid in vessel $$\(P\)$$ is twice as dense as the liquid in vessel $$\(Q\)$$. What is the ratio $$\(\frac{\text { pressure due to the liquid on the base of } P}{\text { pressure due to the liquid on the base of } Q} ?\)$$

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