Let $ABCD$ be a trapezium in which $AB\parallel CD$ and $AB=3CD$. Let $E$ be the midpoint of the diagonal $BD$. If $[ABCD]=n\times [CDE]$, what is the value of $n$? (Here $[\Delta ]$ denotes the area of the geometrical figure $\Delta$).

Let the height of the trapezium be $h$, and let the length of $CD=x$. Since $AB=3CD$, we have $AB=3x$.
The area of the trapezium $ABCD$ is:

Now we find the area of $△CDE$. The area of $△BCD$ has base $CD=x$ and height $h$ (since $CD\parallel AB$ and the distance between them is $h$):

Since $E$ is the midpoint of the diagonal $BD$, in $△BCD$, the segment $CE$ is a median. A median divides a triangle into two triangles of equal area:

We want to find $n$ such that $[ABCD]=n\times [CDE]$:

Therefore, the value of $n$ is $8$.