A container weighs $\frac{7}{8}$ kilograms when it is $\frac{1}{4}$ filled with flour. The same container weighs $1 \frac{7}{10}$ kilogram when it is $\frac{2}{3}$ filled with flour. What is the mass of the empty container?

Step 1 ：Let's denote the weight of the container as C and the weight of the flour as F. We can set up two equations based on the given information.

Step 2 ：The first equation is derived from the first sentence: the weight of the container when it is \(\frac{1}{4}\) filled with flour is \(\frac{7}{8}\) kilograms. This can be represented as: \(C + \frac{1}{4}F = \frac{7}{8}\).

Step 3 ：The second equation is derived from the second sentence: the weight of the container when it is \(\frac{2}{3}\) filled with flour is \(1 \frac{7}{10}\) kilograms. This can be represented as: \(C + \frac{2}{3}F = 1 \frac{7}{10}\).

Step 4 ：We can solve this system of equations to find the values of C and F.

Step 5 ：The solution to the system of equations is: \(C = 0.38\), \(F = 1.98\).

Step 6 ：So, the mass of the empty container is \(\boxed{0.38}\) kilograms.