Two solids $A$ and $ B$ float in water. It is observed that $A$ floats with $\frac{1}{2}$ of its body immersed in water and $ B$ floats with $\frac{1}{4}$ of its volume above the water level. The ratio of the density of $ A$ to that of $B$ is
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(b)Upthrust = weight of body
For $A$ , $\frac{{{V_A}}}{2} \times {\rho _W} \times g = {V_A} \times {\rho _A} \times g \Rightarrow {\rho _A} = \frac{{{\rho _W}}}{2}$
For $ B$, $\frac{3}{4}{V_B} \times {\rho _W} \times g = {V_B} \times {\rho _B} \times g \Rightarrow {\rho _B} = \frac{3}{4}{\rho _W}$
(Since 1/4 of volume of B is above the water surface)
$\frac{{{\rho _A}}}{{{\rho _B}}} = \frac{{{\rho _W}/2}}{{3/4\;{\rho _W}}} = \frac{2}{3}$
For $A$ , $\frac{{{V_A}}}{2} \times {\rho _W} \times g = {V_A} \times {\rho _A} \times g \Rightarrow {\rho _A} = \frac{{{\rho _W}}}{2}$
For $ B$, $\frac{3}{4}{V_B} \times {\rho _W} \times g = {V_B} \times {\rho _B} \times g \Rightarrow {\rho _B} = \frac{3}{4}{\rho _W}$
(Since 1/4 of volume of B is above the water surface)
$\frac{{{\rho _A}}}{{{\rho _B}}} = \frac{{{\rho _W}/2}}{{3/4\;{\rho _W}}} = \frac{2}{3}$
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