The absolute coefficient of expansion of a liquid is 7 times that the volume coefficient of expansion of the vessel. Then the ratio of absolute and apparent expansion of the liquid is

To find the ratio of the absolute coefficient of expansion to the apparent coefficient of expansion of the liquid, let us first define the relationship between these coefficients.

The absolute coefficient of volume expansion of a liquid (${\gamma }_r$) is related to its apparent coefficient of volume expansion (${\gamma }_a$) and the volume coefficient of expansion of the vessel (${\gamma }_g$) by the formula:
${\gamma }_r={\gamma }_a+{\gamma }_g$

According to the problem, the absolute coefficient of expansion of the liquid is 7 times the volume coefficient of expansion of the vessel:
${\gamma }_r=7{\gamma }_g$
This can also be written as:
${\gamma }_g=\frac{{\gamma }_r}{7}$

Now, let us substitute the value of ${\gamma }_g$ into the relation between the expansion coefficients:
${\gamma }_r={\gamma }_a+\frac{{\gamma }_r}{7}$

Rearranging the equation to solve for the apparent coefficient of expansion (${\gamma }_a$):
${\gamma }_a={\gamma }_r-\frac{{\gamma }_r}{7}$
${\gamma }_a={\gamma }_r(1-\frac{1}{7})$
${\gamma }_a=\frac{6}{7}{\gamma }_r$

The question asks for the ratio of the absolute coefficient of expansion (${\gamma }_r$) to the apparent coefficient of expansion (${\gamma }_a$):
$\frac{{\gamma }_r}{{\gamma }_a}=\frac{{\gamma }_r}{\frac{6}{7}{\gamma }_r}=\frac{7}{6}$

Thus, the ratio of absolute and apparent expansion of the liquid is 7/6.