To solve this problem, we will calculate the length of the cuboid based on the given details:

Given:

1. Volume of the cuboid = $700 \, \text{cm}^3$
2. Height of the cuboid = $7 \, \text{cm}$
3. Width of the cuboid = $\frac{1}{4} \times \text{Length}$

We use the formula for the volume of a cuboid:
$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$

Step 1: Represent width in terms of length

Let the Length be $L$. Then,
$\text{Width} = \frac{1}{4}L$

Step 2: Substitute into the volume formula

$700 = L \times \frac{1}{4}L \times 7$

Simplify:
$700 = \frac{7}{4}L^2$

Multiply through by 4 to eliminate the fraction:
$2800 = 7L^2$

Step 3: Solve for $L^2$

$L^2 = \frac{2800}{7} = 400$

Take the square root of both sides:
$L = \sqrt{400} = 20 \, \text{cm}$

Final Answer:

The length of the cuboid is $\mathbf{20 \, \text{cm}}$.

Would you like me to explain further or assist with similar problems? Here are five related questions to explore:

1. How is the relationship between width and length applied in other geometric shapes?
2. How do you derive the volume formula for a cuboid?
3. What would the width of the cuboid be if the length is 20 cm?
4. How do we solve similar problems if the shape is a cylinder or a sphere?
5. What is the importance of units in volume calculations?

Tip: Always check the consistency of dimensions (e.g., cm, m) in your calculations to avoid errors!