Given:
- In triangle ABC, AB = 6 cm
- DE is parallel to BC
- AE = 1/4 AC

To find:
- The length of AD

Approach:
1. Since DE is parallel to BC, we can use the concept of similar triangles to find the length of AD.
2. We will use the property that corresponding sides of similar triangles are proportional.

Solution:

Step 1: Identify the Similar Triangles
- Triangle ABC and triangle ADE are similar because they have a pair of corresponding angles equal and the corresponding sides are parallel.

Step 2: Set up the Proportion
- Let AD = x be the length we need to find.
- Now, according to the property of similar triangles, we can set up a proportion between the corresponding sides.
- In triangle ABC, we have AC/AB = AE/AD.

Step 3: Solve the Proportion
- Substituting the given values, we have AC/6 = (1/4)AC/x.

Step 4: Simplify and Solve for x
- Cross-multiplying the proportion, we get 4(AC) = 6(x).
- Dividing both sides by 6, we have 4/6 = x/AC.
- Simplifying further, we get 2/3 = x/AC.

Step 5: Substitute Known Values and Solve for x
- Since AE = 1/4 AC, we can substitute this value into the equation.
- 2/3 = x/(4AC).
- Cross-multiplying, we get 2(4AC) = 3x.
- Simplifying, we have 8AC = 3x.
- Dividing both sides by 3, we get 8/3 AC = x.

Step 6: Substitute Known Values and Solve for x
- Since AE = 1/4 AC, we can substitute this value into the equation.
- 2/3 = x/(4AC).
- Cross-multiplying, we get 2(4AC) = 3x.
- Simplifying, we have 8AC = 3x.
- Dividing both sides by 3, we get 8/3 AC = x.

Step 7: Substitute the Given Value and Calculate
- Since AC = 6 cm, we can substitute this value into the equation.
- x = 8/3 (6) = 16 cm.

Step 8: Final Answer
- The length of AD is 16 cm.