The extension in a spring is directly proportional to the weight attached to it. If a weight of 5 kg produces an extension of 0.4 cm, what weight will produce an extension of 5 cm?

Introduction / Context:
This is a direct proportion problem based on Hooke's law, which states that, within the elastic limit, the extension in a spring is proportional to the load applied. Such questions are common in quantitative aptitude because they test your ability to use proportional reasoning with very little calculation.

Given Data / Assumptions:
• Extension in the spring is directly proportional to the weight attached. • A weight of 5 kg produces an extension of 0.4 cm. • We want the weight that will produce an extension of 5 cm.

Concept / Approach:
If one quantity is directly proportional to another, we can write W1 / x1 = W2 / x2, where W1 and W2 are the weights and x1 and x2 are the corresponding extensions. Equivalently, W = k * x for some constant k. First, we find the constant of proportionality using the given data, and then apply it to the required extension. It is important to keep units consistent, but here both extensions are in centimetres, so we can work directly.

Step-by-Step Solution:
1. Let W be the weight in kilograms and x be the extension in centimetres. 2. Given the direct proportionality, W = k * x for some constant k. 3. From the given data, when W1 = 5 kg, x1 = 0.4 cm, so 5 = k * 0.4. 4. Solve for k: k = 5 / 0.4. 5. Compute 5 / 0.4 = 5 / (4 / 10) = (5 * 10) / 4 = 50 / 4 = 12.5. 6. Therefore, k = 12.5 kg per centimetre. 7. For an extension x2 = 5 cm, the required weight W2 = k * x2. 8. Substitute: W2 = 12.5 * 5 = 62.5 kg.

Verification / Alternative check:
You can verify by using the ratio form of direct proportion. Since W1 / x1 = W2 / x2, we have 5 / 0.4 = W2 / 5. Therefore, W2 = 5 * (5 / 0.4) = 25 / 0.4 = 62.5 kg. This matches the value we obtained using the constant k method. Both methods show consistent results, confirming the correctness of the calculation.

Why Other Options Are Wrong:
• 6.25 kg: This would correspond to a much smaller extension than 5 cm when using the same proportionality constant. • 4 kg: This is less than the original 5 kg, yet the extension desired is much larger than 0.4 cm, so it cannot be correct. • 40 kg: This gives W2 / x2 = 40 / 5 = 8, which does not match 5 / 0.4 = 12.5. • 12.5 kg: This corresponds to an extension of 1 cm, not 5 cm, under the same proportionality constant.

Common Pitfalls:
A frequent mistake is to treat the relationship as inverse proportion by accident, or to mis-handle the division by a decimal such as 0.4. Another pitfall is forgetting to keep both extensions in the same unit, but here that is already consistent. Writing the proportion clearly before substituting numbers helps avoid errors.

Final Answer:
The required weight is 62.5 kg.