The length, breadth and height of a cuboid are in the ratio 19 : 11 : 13. If the length is 30 cm more than the height, what is the volume of the cuboid in cubic centimetres?

Introduction / Context:
This problem combines ratio of dimensions with an additional linear condition to determine actual dimensions of a cuboid. Once the real length, breadth and height are found, the volume can be computed by simple multiplication. Such questions are common in aptitude tests to check whether you can move from proportional relationships to specific values using extra conditions.

Given Data / Assumptions:

- Length : breadth : height of a cuboid are in the ratio 19 : 11 : 13.
- Let the common multiplying factor be x centimetres.
- So length = 19x, breadth = 11x and height = 13x.
- It is given that length is 30 cm more than height.
- We need to find the volume of the cuboid in cubic centimetres.

Concept / Approach:
When dimensions are in a given ratio, you can represent them as multiples of a common variable. An additional equation, such as a fixed difference between two dimensions, allows you to solve for this variable. Once the exact numerical values of length, breadth and height are known, the volume is given by V = length * breadth * height. Careful substitution and multiplication lead to the final volume.

Step-by-Step Solution:
Step 1: Represent the dimensions using the ratio and a common factor x.Step 2: Length L = 19x, breadth B = 11x, height H = 13x.Step 3: Given that length is 30 cm more than height, so L = H + 30.Step 4: Substitute: 19x = 13x + 30.Step 5: Simplify: 19x - 13x = 30 which gives 6x = 30.Step 6: Therefore, x = 30 / 6 = 5.Step 7: Compute actual dimensions: L = 19 * 5 = 95 cm, B = 11 * 5 = 55 cm, H = 13 * 5 = 65 cm.Step 8: Volume V = L * B * H = 95 * 55 * 65 cubic centimetres.Step 9: First 95 * 55 = 5225, then 5225 * 65 = 339,625 cubic centimetres.

Verification / Alternative check:
Check that the difference between length and height equals 30 cm. With L = 95 cm and H = 65 cm, L - H = 95 - 65 = 30 cm, which matches the given condition. Also, ratios of 95 : 55 : 65 reduce by dividing each term by 5 to 19 : 11 : 13, confirming that the solution respects the original ratio. Thus the computed volume 339,625 cubic centimetres is consistent with all given information.

Why Other Options Are Wrong:
Volumes such as 81,510 or 89,665 cubic centimetres correspond to incorrect values of x or arithmetic errors when multiplying the dimensions. 195,300 and 410,000 cubic centimetres are far from the calculated value and usually result from misreading the ratio or forgetting the height condition. None of these options satisfy both the ratio and the 30 cm difference requirement simultaneously.

Common Pitfalls:
Students sometimes misinterpret the ratio as length : height : breadth or rearrange the order, leading to wrong dimension assignments. Another common mistake is to subtract the wrong way when applying the condition L = H + 30, or to miscalculate products like 95 * 55. Always keep the ratio order consistent, handle the linear equation carefully and perform multiplications step by step to avoid errors.

Final Answer:
The volume of the cuboid is 339,625 cubic centimetres.