A line of true length 398 m, when measured by a 20 m chain, is recorded to be 400 m. What is the actual length of the chain (in m)?

📝 Detailed Explanation: Correction for Incorrect Chain Length

This problem deals with a systematic error caused by using a chain that is not of its designated standard length. We can solve this by applying a fundamental surveying principle.

Principle of Incorrect Chain Measurement

The core idea is that the true total length measured is always constant. This can be expressed with the formula:

True Length of Line × True Length of Chain = Measured Length of Line × Incorrect Length of Chain

However, a more intuitive formula is:

$$ \text{True Distance} = \frac{\text{Actual (Incorrect) Length of Chain}}{\text{Designated Length of Chain}} \times \text{Measured Distance} $$

Calculation Steps

Let's identify the given values:

• True Distance (l) = 398 m
• Measured Distance (l') = 400 m
• Designated Length of Chain (L) = 20 m
• Actual Length of Chain (L') = ? (This is what we need to find)

Now, we substitute these values into the formula:

$$ l = \frac{L'}{L} \times l' $$

$$ 398 = \frac{L'}{20} \times 400 $$

To solve for L', we rearrange the equation:

$$ L' = \frac{398 \times 20}{400} $$

$$ L' = \frac{7960}{400} $$

$$ L' = 19.9 \text{ m} $$

Therefore, the actual length of the chain is 19.9 m. Because the chain was shorter than its standard 20 m length, more chain lengths were needed to cover the distance, resulting in a measured distance (400 m) that was longer than the true distance (398 m).