In triangle ABC, DE is parallel to AB. If DE is one-fifth AB, what is the ratio of the area of triangle DEC to the area of triangle ABC?

for any similar figures the ratio of areas is the square of the ratio of lengths.

so
1/25

You can easily prove this for yourself by noting that if DE = AB/4
Then the altitude of the little triangle = 1/5 the altitude of the big triangle
(1/2)(1/5)(1/5) = (1/25) * (1/2) (1)(1)

by the way, for similar SOLID 3 d figures:

areas are proportional to square of length ratio
Volumes are proportional to CUBE of scale ratio
This is handy to use for quick estimates
For example compare a 500 foot long oil tanker to a similar 1000 foot tanker
the big one has 2^2 = 4 times the surface area in the water
however it is 2^3 = 8 times the volume and therefore cargo capacity.
The water resistance is roughly proportional to area
so it carries 8 times the oil with 4 times the drag
so it carries twice as much oil per horsepower for the same speed.
That is why ships keep getting bigger and bigger.

You can easily prove this for yourself by noting that if DE = AB/"5" **not four**

To find the ratio of the areas of triangle DEC to triangle ABC, we need to find the corresponding sides of the two triangles.

Since DE is parallel to AB, we can determine that triangle DEC and triangle ABC are similar triangles. This means that their corresponding sides are proportional.

Given that DE is one-fifth of AB, we can conclude that the ratio of the corresponding sides of triangle DEC to triangle ABC is also 1:5.

Since the area of a triangle is proportional to the square of its corresponding sides, we can determine the ratio of the areas.

The ratio of the areas of triangle DEC to triangle ABC is equal to the square of the ratio of their corresponding sides. In this case, the ratio is 1:5, so the square of this ratio is (1/5)^2 = 1/25.

Hence, the ratio of the area of triangle DEC to the area of triangle ABC is 1/25.