Let the total cost in January = t2
Let the total cost before January = t1
Let the fixed cost in January = f2
Let the fixed cost before January = f1
Let the variable cost in January = v2
Let the variable cost before January = v1
Given that t2 = f2 + v2, and t1 = f1 + v1, by what percent did the total cost of producing item Y change in January?
$$\frac{t2}{t1}=\frac{f2+v2}{f1+v1};\ where\ v2=\left(100\%-10\%\right)v1$$
$$v2=90\%v1=0.9v1$$
$$\frac{t2}{t1}=\frac{f2+0.9v1}{f1+v1}$$

Statement 1: The fixed cost of producing item Y increased by 12% in January.
f2 = (100% + 12%)f1 = 112%f1 = 1.12f1
$$\frac{t2}{t1}=\frac{f2+0.9v1}{f1+v1}$$
$$=\frac{1.12f1+0.91v1}{f1+v1}$$
The information provided is not enough to get the exact value of % change in the total cost of producing item Y. Therefore, statement 1 is NOT SUFFICIENT.

Statement 2: Before the changes in January, the fixed cost of producing item Y was 5 times the variable cost of producing item Y.
f1=5v1
$$Hence,\frac{t2}{t1}=\frac{f2+0.9v1}{5v1+v1}$$
The information provided is not enough to get the exact value of % change in the total cost of producing item Y. Therefore, statement 2 is NOT SUFFICIENT.

Combining both statements together:
$$\frac{t2}{t1}=\frac{f2+0.91v1}{f1+v1};\ \ \ \ \ \ where\ f2=1.12f1,\ and\ \ \ f1=5v1$$
$$Therefore,\ f2=1.12\left(5v1\right)=5.6v1$$
$$\frac{t2}{t1}=\frac{5.6v1+0.91v1}{5v1+v1}$$
$$\frac{t2}{t1}=\frac{6.5v1}{6v1}$$
$$=\frac{6.5}{6}$$
=1.08

% change = (1.08 * 100)% - 100%
= 108% - 100% = 8%

From the expression above, it's obtained that the cost of producing item Y increased by 8% in January. Therefore, both statements combined together are SUFFICIENT.

Answer = option C

Cheers!
Deloitte