Question

A body is found to be floating in water with one fourth of its volume above the surface of water. If the body is kept in oil, it floats with one third of its volume above the surface of oil. The density of oil can be given as,
$\begin{aligned} & A.\dfrac{3}{4} \\\ & B.\dfrac{4}{9} \\\ & C.\dfrac{2}{3} \\\ & D.\dfrac{9}{8} \\\ \end{aligned}$

Answer 1

The volume of the body above water is one -fourth. Therefore the volume of the body inside the water will be ${{\dfrac{3}{4}}^{th}}$ of its volume. In oil it is mentioned that the body is placed one third above the oil. Therefore the two-third of the body is inside the oil. This information will help you in solving this question.

Complete step by step answer:
As mentioned in the question, the volume of the body above water is one -fourth. Therefore the volume inside the water will be ${{\dfrac{3}{4}}^{th}}$ of its volume.
Let us assume that the volume of the body be $V$ and density be $d$.

Then in the case of water, we can write weight of the body as,
$V\times d\times g=\dfrac{3}{4}V\times {{d}_{water}}\times g$
Here $g$ be the acceleration due to gravity.
Now let us analyse the case of oil.
The one-third of the body is mentioned in the question to be above the oil. Hence we can say that the volume of the body present inside oil will be ${{\dfrac{2}{3}}^{rd}}$ of its volume.

Therefore we can write that,
$V\times d\times g=\dfrac{2}{3}V\times {{d}_{oil}}\times g$
Where ${{d}_{oil}}$ be the density of the oil.
As the left hand side of both the equations are equal, we can write that the right hands will also be equal. That is,
$\dfrac{3}{4}V\times {{d}_{water}}\times g=\dfrac{2}{3}V\times {{d}_{oil}}\times g$
Cancelling the common terms in both the equation will give,
$\dfrac{3}{4}V\times {{d}_{water}}=\dfrac{2}{3}V\times {{d}_{oil}}$
Rearranging the equation can be written as,
$\dfrac{\dfrac{3}{4}}{\dfrac{2}{3}}=\dfrac{{{d}_{oil}}}{{{d}_{water}}}$
That is,
$\dfrac{{{d}_{oil}}}{{{d}_{water}}}=\dfrac{9}{8}$
Hence the answer for the question has been calculated. The answer is given as option D.

Note:
The density is defined as the mass per unit volume. Density is a scalar quantity which depends only on the magnitude of the quantity not the direction of the body. Its unit is given as kilogram per cubic metre.